JPH06503194A - Object image creation system in alternating geometry - Google Patents

Abstract

(57) [Summary] This bulletin contains application data before electronic filing, so abstract data is not recorded.

Description

[Detailed description of the invention] Technical field of invention of object image creation system in alternating geometry The present invention relates to the field of image production devices, for educational, scientific, commercial and/or entertainment applications. It is used in the field.

Background technology of the invention Since the mid-19th century, the field of geometry has been based on fractions other than Euclid's fractions. He has shown an interest in geometry. numbers like Lobachevsky and Riemann Scholars have considered the possibility of relaxing Euclidean fractions. Euclid, which has received particular attention. [Parallel line segment ratio J of [If a given point is not on a given straight line] If the given line passes through that point, there is only one straight line that does not intersect the given line. "No."

To relax the parallel fraction, any number of straight lines passing through points outside the given straight line are given. Prove that it does not intersect with the given straight line, or prove that none of those straight lines intersect. We have to prove that this kind of relaxation does not apply to hyperbolic geometry or curved spaces. It will be produced. Euclid's "parallel line fraction".

It was found that such a transformation of is a space in which the tllI structure has changed dramatically. .

For example, Lobachevsky's geometry is hyperbolic geometry, in which the dimensions of a shape are To change the law, you must also change its shape, which means that the length has an absolute This means that there are standards.

Also, numbers that apply commonly to various geometries in dimensions greater than three dimensions grow. It's here. And over the past century, there has been extensive speculation about a "fourth dimension." It has been done. Besides metric space geometry, non-metric space geometry has fully grown. non-distance Distant geometry does not assume the concept of length, and is generally a has a wide-ranging theory and has systematized the theory in the area of topology. . Differential geometry made it possible to construct a theory of curved spaces.

In one application of four-dimensional geometry, the fourth dimension is interpreted as time.

Einstein's special theory of relativity assumes that the fourth dimension is time. I made it so that it would be accepted as Eno's. On the other hand, the curvature of physical space or spatial temporality The continuum between The concept has changed to that of a connoisseur. 17 However, this has nothing to do with geometry itself. Not only. There exists a fourth spatial coordinate for describing space. .

Henri Poincaré explains our intuitive geometrical concepts and the solid objects we are familiar with. It has been pointed out that there is a close relationship between @mai. Especially the human eye is 3D Think of it as a solid object that moves with the motion of a solid object in Euclidean geometry. I can do it. Mathematicians and others generally agree that our spatial intuitions are like this [7 Formed and limited according to three-dimensional space due to growth development, genetics, or human nature It is believed that the temperature is at °C. formally or intellectually in alternating geometry Although we can understand various shapes and relationships, we cannot directly visualize these shapes and relationships. It is generally accepted that it is impossible to do so. However, only many numbers It is useful for students, teachers, engineers, and other professionals who use applied numbers. It is widely recognized that ``spatial intuition''13 can sometimes be a great aid to mathematical understanding. being admired.

"Spatial intuition" is something that the human visual system has had since ancient times. ), when presented with 2D information, accept it as 3D information attached to it, It is the ability to presume the existence of three dimensions as a matter of course.

As you can see, we can use our insight into one dimension to For example, the web of our eyes The membrane is only two-dimensional, yet it can sense three-dimensional objects. A series of two-dimensional images are formed one after another on the retina, and then we see a three-dimensional object. This is because we intuit it by expanding its depth. We unconsciously move our heads and do things. It uses technology that determines the angle of the body relative to the background. 6 In such a case, each two-dimensional image on the retina of our eye actually It is placed in a relative position, such as behind, in front, to the side, above, etc., with respect to the line of sight axis.

Consider the lack of object sensing ability in alternating geometry such as 4-dimensional Euclidean geometry. This lack of intuition is considered to be an important handicap. Many methods have been considered to compensate for this. In these methods, the representation of the object is based on alternating geometry, e.g. It is done within four-dimensional Euclidean geometry.

In that case, customary spatial or three-dimensional structures such as projections, intersections, or models may be used. The method of lid geometry is used in a number of forms. Generally speaking, these The method uses some kind of transformation to Image the shape formally defined in surrogate geometry as a shape in three-dimensional Euclidean space It is done by Only in this way can we examine the form in the usual way. This is possible. In the simplest case, such a shape is a cubic form of a shape in alternating geometry. It may be an intersection with the original Euclidean space, but it becomes complicated. If so, it is a three-dimensional model of the solid, and is In this way it may be something that can be rotated in the shadow. , the form and complexity of the original shape can be rotated in the original space, the mode of projection or the center by changing it and observing the effect in the resulting object. Line diagrams and models of this kind may appear over the years, and may be used to discuss alternating geometry. It has been a great help to me in my discussion7 With the advent of electronic meters, the ability to visualize alternating geometry was greatly enhanced, Images are formed and transformed at high speed according to instructions. Also, various hypothetical lighting modes The changes in the original color of the do and the changes in the animation are expressed.

However, these images are still projections of objects in three-dimensional Euclidean space. is an intersection, a model, and appears within the alternating geometry itself. You need to remember that it is different from your body. 4-dimensional Euclidean geometric characters like this The projections, intersections, and models of such objects are examples of how they have been displayed on computer systems. By Thomas Banchoff Dimensio n: Geome, Com ufer Gra hics and Hi he rDimension (1990) and ■ The film by The Hercube: Pro’ection and SIicin (1978), in which Mr. Banchoff uses 4D space. Define a 4D object like a hypercube inside a hole in normal space. Intersect images and calculate and display them as orthogonal or stereographic projections. ing. In orthogonal projection, the normal coordinates are projected as they are without any change, and the fourth coordinate (corresponding to the projection from a light source at an infinite distance) is ignored. stereograph In Fickian projection, a ray or straight line emanating from a point at an infinite distance creates a third-dimensional image. The original object is generated. The image thus generated is a 4D cube or Produce anis movie of Ipercube and 4D Sphere or Hypersphere In Nira's movies, which are used to rotate objects, Dramatic Anis movement in full color with computer-enhanced light and shadows is produced by the effect of The images produced greatly enhance intuitive sensing of four-dimensional objects. I'm taking lots of pictures. However, in these images, from objects in alternating geometry It does not reproduce the effect of directly emitted light. Therefore, the 4D object does not show images of himself. It only shows the projection of a 4D object into 3D. be.

Alternating geometry has a wide range of applications. Within mathematics, for example, place A fourth dimension is always required when a plane is imaged onto itself by means of a correlation number. Also is the image of one pair of numbers to another, as in the case of drawing the graph of a function of a complex variable. The same applies to the case. Physicists and engineers often use systems with more than three variables. There are times when I do some work. In general, the configuration space of a system with more than three variables is It is better to use 0 curve coordinates, which create a trajectory with higher dimensionality when drawing a graph. Get results. It is often best to express nonlinear systems using curved coordinates. However, to visualize the curvature of a three-dimensional space it is It is necessary to embed it in the original linear space. Graphical design and image representation Today, a wealth of computer technology is available for various modes of virtualization. Real-life experiences are also possible and can be used to enhance visual intuition of objects in alternating geometry. Isn't there a need for some type of equipment that is a projection and model of conventional technology? This gives the effect of directly visualizing the object itself in the alternating geometry, rather than the intersection. It seems necessary to

Disclosure of invention The purpose of the invention is to directly visualize objects in alternating geometry and/or to The objective is to provide a system for the movement and projection of objects in alternating geometry space. . Examples of such alternating geometry are 4-dimensional Euclidean geometry and 3-dimensional bee. Examples include Evsky geometric characters, but are not limited to these.

A further object of the invention is to solve the problem by modeling, intersecting or projecting objects in alternating geometry. The purpose of the present invention is to provide a system that allows direct visualization of objects without having to use them.

A further object of the present invention is to directly visualize objects in four-dimensional Euclidean geometry. The goal is to provide a system that can help.

The next objective of the present invention is to directly visualize objects in 3D Labochevsky geometry. The goal is to provide a system that can help.

The next object of the present invention is to provide a system for direct visualization of objects, which also allows interaction. Visualization and manipulation of visualized images of objects in alternating geometry ) is to provide a system that makes it possible.

The next object of the present invention is to provide a system for direct visualization of objects, which also allows interaction. Visualization and manipulation of visualized images of objects, especially in 4D unified geometry. The goal is to provide a system that allows you to do what you want.

The next object of the present invention is to provide a system for direct visualization of objects, which also allows interaction. Visualization and manipulation of visualized images of objects, especially in 3D Lobatievsky geometry The aim is to provide a system that allows this to be done.

The next object of the present invention is to enable transformation of objects in alternating geometry and rotation on a plane. The objective is to provide a system that enables

The next object of the present invention is especially to transform objects in 4D Euclidean geometry and to 6. The next object of the present invention is to provide a system that allows rotation in the In particular, we consider the transformation of objects in 3D Lobatievsky geometry and the symmetry on the plane. The goal is to provide a system that makes it possible.

The next purpose of the present invention is to observe an object in geometry that alternates rotation and translation motion on a plane. The goal is to provide a system that allows attribution to users.

A further object of the invention is to provide a system that allows imaging of objects in alternating geometry. It is to be.

A further object of the invention is to specifically enable the imaging of objects in 4D Euclidean geometry. The objective is to provide a system that allows

The object of the present invention is to improve the geometric construction or drawing of objects in alternating geometry. The goal is to provide a system that makes it possible.

A further object of the invention is in particular the geometric configuration of objects in 3D Lobatievsky geometry. Or to provide a system that enables drawing.

The next object of the present invention is to understand objects in alternating geometry and how they appear in the space of alternating geometry. This means that it includes a system that images the image as it appears. In this case, alternating geometry has Includes but is limited to 4D Euclidean geometry and 3D Lobachievsky geometry It is not something that will be done.

These objectives are achieved by following the systems and methods of the present invention.

The present invention is a system that forms an image of an object in an alternating geometry on the retina of the user's eye. stem and method. Alternating geometry and objects are input to the system. be done. The transducer detects objects forming objects in alternating geometry. Convert object points to corresponding 2D image points in 3D uniclid space. vinegar Then, the system's processing unit assembles the transformed image points and places them in the alternating geometry. The image should be of an object that is directly visible to the user's eyes. output from the processing device The input information controls the output device to compose the objects that the user sees. The assembled image points are presented.

More specifically, the image forming system and method of the present invention are based on 4D Euclidean and Objects in alternating geometry such as 3D Lobachievsky are Performs a visual display. Moreover, this imaging system and method How to interactively change the user's perspective during rotation and translation and/or alternating geometry This provides a fully interactive imaging system.

Furthermore, the transducer uses the geometric relationships of the alternating geometry to A ray from an object point enters the 3D unified space of the user's eye. The X and Y axes are used as references to determine the angle at which the X and Y axes should be used.

The transducer then uses the resulting angle of incidence to direct the output device and the user's eye. Determine the equivalent angle in 3D space. In this case, each image point on the retina The underlying 3D Euclidean geometric relationships are used to determine the coordinates. compilation The computer collects and assembles the image points, sends them to a computer monitor, and analyzes them in alternating geometry. Allows objects to be displayed as they appear exactly as they appear.

Brief description of the drawing For the objects already mentioned and other objects, forms, and examples of the present invention, please refer to the following drawings. This will be explained in more detail. FIG. 1 here illustrates an example of a 4D hypercube. Ru.

FIG. 2 illustrates the 3D unified space of the human eye.

Figure 3 shows that a ray from a 4D point outside a given 3D space is a 3D space. It illustrates the fact that they can only intersect at one point.

Figure 4 illustrates the rotation of a shoe in 4D space as seen in 3D unified space. Ru.

Figures 5(a) and 5(b) each show "F" rotating in 3D unified space. The view of the character as it is at a 2D station without any help, and the view of the same object with the aid of the present invention. 8 Showing the view taken from a distance FIG. 6 shows the 3D unified space with eyes of the transducer system of the present invention. (usual space) and 4I) Figure 7 shows a point P in Euclidean space. The human eye sees a 3D object from its position and direction without any help. shows.

Figures 8(a)-8(d) show the 2D eyes detecting 2D objects in 3D space. Figure 3 illustrates what is viewed using the system of the present invention.

FIG. 9 is a flowchart illustrating all functions of the system of the present invention.

FIG. 1O is a flowchart illustrating, step by step, the general functionality of the system of the present invention. It is.

Figure 10(a) shows an example of a device according to the invention.

Figure 11 shows how one ray from one object point in 4D space is generated by the system of the present invention. 3 illustrates geometric relationships that are transformed into 3D space.

FIG. 11(a) shows that the transducer of the present invention projects a light beam onto the user's eye; The effect is to form an image of an object in an alternating space on the retina of the user's eye. It also shows how the display device displays the image as it appears to the human eye. Showing the offset effect of the screen.

Figures 12-14 show that a single beam of light produces an image on a screen used by the transducer of the present invention. Figure 3 illustrates the geometric relationships in 3D space seen in forming points.

FIG. 15 shows the “Interactive Visualization and Operation” of the present invention. ALIZATION AND MANIput, AnoN') mode Go/View/Touch (touch) J (Go 几00に7rOUCH) mode controller Troll panel and screen display shown.

Figures 16(a)-16(d) illustrate the ``Interactive Visualization and Manipulation'' ('IN''rE RAC″rIVE VISUALIZATION AND MANIPULATI “Go/Look/Touch J” (Go/LOOK/TOU) in ON’) mode CH) and “Move/Turn/Touch J (MOVE/TURN/TOUCH) mode” The figure shows each functional module of the control panel for the board.

Figure 17 shows the [Go/Look/Touch J (Go/LOOK/rOUcH) mode] “Touch (Touch) J (TOUCH) test flowchart is illustrated.

18(a) to 18(d) illustrate the “Interactive Visualization and Manipulation J” (’INTE) of the present invention. RACTIVE VISUALIZATION AND MANIPULATION Go/LOOK/TOU in N’) mode CH) A diagram illustrating the screen display and panel controls by way of example mode. be.

Figures 19(a)-19(c) show the 3D Lobatsky ski sky created by the system of the present invention. Geometric relationship when a ray from an object point in between is transformed into 3D space Illustrating the person in charge.

FIG. 19(d) shows that a ray from the 3D Lobatsky space is transformed into the transducer of the present invention. 1- Geometric relationships in 3D space when forming one image point on the screen used by the server Illustrated.

Figures 20(a)-20(h) are 4[) to draw a point in Euclidean space. ``Drawing/Image Formation J (GRAPHINO/IMAOING) Mode'' of the present invention Figure 3 illustrates the screen display and panel controls for.

Figures 21(a)-21(d) are visualizations of coordinate grid lines (decals in 4D space). The graphing/image forming method of the present invention can be used to plot a function with four variables corresponding to ” (Illustrating the case of performing according to GRAPHIN (3/IMAGING) mode Ru.

22(a)-22(d) show the graphing of complex numbers in 4D according to the present invention. Follow the formatting/image forming J (ORAPHING/IMAGING) mode. The following is an illustration of the case.

Figures 23(a)-23(d) show the graphing of hyperbolic cosine in 4D according to the present invention. According to the “Graphing/Imaging” mode (GRAPHINO/IMAGING) of The following figure shows a case in which this is done.

Figures 24(a)-24(d) show the images in Figure η at 60 degrees in 4D. , 120 degrees, 150 degrees, and 180 degrees of rotation are illustrated.

Figures 25(a) to 25(e) are planes in any 4D direction for the graph in Figure 24. Figure 3 illustrates a partitioning showing the data structure of.

Figure 26 receives laboratory input from two resonant systems connected to each other. 1 illustrates a HYPERGRAPH device;

Figures 27(a)-27(e) geometrically construct objects +- in alternating geometry. Virtual blackboard screen in BLACKBOARD mode of the present invention for Showing display and panel controls.

Figures 28(a)-28(c) show the geometric objects of alternating geometry on the Euclidean blackboard. Provides a screen display of problems when configuring a project.

Figures 29(a)-29(b) show the 3D robot in the blackboard mode of the system of the present invention. Creation of Lambert's quadrilateral in Bachikowsky geometry and on the Euclidean blackboard The screen display when the test was performed and when the test was performed on the Lobachevsky blackboard are shown below. Shown below.

Figure 30 is a label with each side of X, Y, Z, and m in 3D unified space. This shows a quadrilateral.

Figures 31(a)-31(b) show Lovatache in BLACKBOARD mode. This figure illustrates the case of drawing a straight line passing through two points P and Q on a ski blackboard.

Figure 32 shows Figures 31(a)-31(b) in the blackboard mode of the system of the present invention. 2 is a flowchart of the procedure of a system for drawing a diagram.

Figures 33(a)-33(d) are the drawings of Figures 31(a)-31(b) in blackboard mode. shows the case where the procedure is performed according to the flowchart in Fig. 32, and the 3D Euclidean Corresponding results when performed on a blackboard are also shown.

BEST MODE FOR CARRYING OUT THE INVENTION For a thorough understanding of the concepts related to the present invention, please refer to Figures 1-8 and the following general introduction. After mentioning this, I will move on to a detailed description of the invention. Also, it was first released in 1984. Flatland, written by Abbott, published by E, /L (Slgnet C1assic, 1984) also lives in the world of W, has 2D eyes, As a background for intuitively understanding the hypothetical case of people with one-dimensional retinas. I'll mention it.

In order to perceive higher-dimensional objects, and (this is more important) A basic understanding of geometric objects is required to understand the capabilities of inventive optical systems. It is. Geometric shapes exist in dimensionality and depend on the definition of the space in which they are defined. It exists according to principles and proportions. Therefore, the hypothetical 2D-Euclidean space or In the world (that is, a plane), the typical shape is a square, whereas the cubic shape is In the original Euclidean space, a cube is representative. On the other hand, Lobachevsky's geometry A square is an unthinkable shape.

When we think about the problem of dimensionality, we consider objects in one dimension and objects in one dimension. Starting from the geometric relationships between objects of higher dimensions, objects of a higher dimension than our own You can move towards understanding the shape of your body. From a broader perspective, I think of a single “point.” Starting from a dimensionless object, we can trace geometric continuity and create a hypercube. It is also possible to try to reach objects of 4D Euclidean geometry, such as curves. come. A point in the 0th dimension is extended by a unit distance to a new dimension called the 1st dimension, which is called a line. Things can be created. This shows the new degree of freedom that one point has gained.

Similarly, in 2D space, a straight line can move perpendicular to its length by a unit distance. A square is created.

In the case of three dimensions, if a square moves a unit distance perpendicular to its plane, a cube is created. . Proceeding further to four-dimensional space, if the cube moves by a unit distance from its volume, the first A 4D object called a hypercube (created according to the invention) shown in the figure is formed. Ru. Geometric continuity means that the hypercube in Figure 1 has 16 vertex angles. On the other hand, this can be understood by looking at the fact that a cube has an apex angle of 8.

The term "space" is not limited to the literal meaning used in this discussion; ``color space'', ``tone space'', ``configuration'' used for currently existing systems. As in the case of ``space'', it should be understood both as food and in an abstract sense. be. In this way, the present invention allows the visual images it produces to It can be similarly applied to formal and melancholy spaces.

Figure 2 shows the human eye trapped in a 3D unified space. . The eye shown in Figure 2 has a plane π (retina) and a bright pupil located outside that plane, 0). It is limited to three dimensions as an object. The eye is shown as space S0 ing. The spatial formula is illustrated as a box, but it can actually be seen in Figure 3. It does not have a "ridge". In general, humans use normal spatial equations with eyes, that is, 3D units. It is believed to be trapped in Tsudo space. Therefore, the 3D uniformity of the eyes There is a problem with directly looking at objects outside of Tsudo space. 3D unique space A ray of light emitted from a point P on an object in alternating geometry outside of is Even if you intersect with this space, it will only be once, and you will never be able to repeat it again. It is possible to prove mathematically that it will not come.

In Figure 2, it is a normal space S. A point P in the space of alternating geometry is shown outside of . So Now, as shown in Figure 3, when a ray of light emitted into the outer space of alternating geometry intersects with the pupil, If this happens, the light rays will not be able to intersect with the retina and will not be visible. now This logic was accepted as completely valid in the mathematical community to this day, and 4D Objects in alternating geometry, such as objects in cliched space, have their properties logically understood. , can be modeled using various methods in ordinary Euclidean space. However, it was never thought possible to see it. This way In this way, prior to the present invention, it was not possible for people to see a direct projection of a 4D object. The present invention overcomes this problem by using a transducer. Ta. This will be discussed below.

Anne 9 Poincaré says that experienced mathematicians practice 4D Euclidean 1; He said that it should be possible to see the phenomena between them with the naked eye. 4D Yukuri In Tsudo space, an object pointing to the left is always rotated to become an object pointing to the right. say This means that the shoe for the left foot is rotated to become the shoe for the right foot. This is the first The 4 spatial coordinates provide an additional degree of freedom. In Figure 4, shoes in 0 space A view of the rotation in normal space is shown. as shown here , Shoe 1 for the left foot does not appear to be rotating, but gradually shrinks and eventually disappears. Put it away. However, it reappears and grows into a life-sized shoe for the right foot.

Here, the problem that the present invention overcomes is that it is not a fleeting 3D image as described above, but a 4D object. The goal is to be able to clearly observe the entire picture from the beginning to the end of the movement. This is temporary This can be more easily understood by referring to Figure 5(a), which shows a fixed 2D world (i.e., a plane). Wear.

As shown here, humans in such a 2D space are 2D and have 2D eyes and It has a one-dimensional retina. In Figure 5(a), no help is received from outside 2D. Eyes turned to the left (showing where you saw the “F” in Ievular rotate) Ru.

In 2D's eyes, the left-facing r F J appeared, then began to shrink and then disappeared. Then, it turns right again [dextral's [FJ] appears. However However, as shown in Figure 5 [b1], if the 2D eyes are If you can look at it with a helper, it means that [F] is actually in 3D space (that is, in normal It rotates in space), does not change its dimensions, but simply rotates 180 degrees (that is, turns inside out). You should see them doing something (becoming something). 3D space has degrees of freedom in rl-” shape. Due to the extra weight, it disappears from the 2D world or spatial plane by rotation. There is. This is the same as the shoes in Figure 4 disappearing from the normal space s0 to the 4D space. The same thing.

By using the present invention as described below, it is possible to see the rotation of the shoe and create a 3D object. You will be able to move and rotate your body in 4D space. Conventional technology 3D Compared to projection, model, and intersection techniques, the present invention allows human 3D eyes to 6 The present invention also provides a 3D robot that enables 4D objects to be viewed virtually in situ. It can be extended to any alternating geometry, such as key geometries.

How the invention enables direct viewing of e.g. 4D objects as described above It is important to explain. Visual data from the 4D world enters the human 2D eyes When the images are packed into a large number of images, superimposition and apparent visual confusion inevitably occur. all Not all objects are in the direction in which they were first seen. In that sense These images are not "direct". They appear to be refracted into the eye. . This is similar to how a stick inserted into a pond appears to bend under the water's surface.

Even in such a case, the fact that the stick is not broken means that you can move your eyes to an appropriate position and look at the entire stick. If you look at it for a long time, you will be able to confirm that In either case, rotate the object so that you can look at it directly, and then Confirm by decreasing the distance between the pupil and the 4D object toward zero. In this process Objects can not only be seen directly, but also approached and touched. . This feature will be explained more specifically below. ] and "Move/Turn" (built into a mode called MOVE/TURNI) .

The transducer used in this invention replaces the pupil of the human eye with a "system eye". The normal space S of the eye. and an alternation in which an object is inserted It functions as an interface between the geometric spaces S.

The transducer is adapted to the user's eye and alternating geometry space, effectively alternating geometry. This makes it possible to insert it into the academic field.

-i-wise, the transducer has two functions. First of all, it is an object The angle at which the ray from the target enters the pupil of the system eye relative to the reference axis of the system eye. to be determined. In this case the trigonometry laws of alternating geometry of the space S, are used. next , an ordinary space S that is a 3D Euclidean geometry. According to the law of S is the eye space S of the system from the angle of incidence. Determine the position of the image point in , and then The user's eyes are determined accordingly.

For example, in Figure 6, the normal eye space of the transducer system is designated as Akira. There is. Space S. The point P outside is the space S of alternating geometry, and the 0 point α inside is the pupil of the eye. , and the plane π is #IM. The straight line α looks at the object through the transducer. The rays represent the line of sight axis of the eye in the space S1, which lies in the space of alternating geometry. , enters the transducer or the pupil of the eye at a point α. transducer Determine the angles α and β between the eye system X and the normal space S where the eye is located. Inside generate a new ray ~ at a determined angle to the two axes. ■Is an eye belongs to the space of , and is incident on the retina to form a 2D image point Q. In this way, The suc- cuer is an image corresponding to an object point in an arbitrary alternating geometry space S1. The image point is in normal space S. Generate inside. The rays (a) and (layers are in two spaces If the angles are equal, they appear to be the same line, but they are not the same line. It's important to remember. Also, the angles are equal depending on the type of display used. It is not necessary that they be equal. This optical transducer Only after each use can an image point Q be obtained on the retina. space sl If it is a 4D space, each object point P (x, y , z, v) into image points Q, we obtain Form an image of the object on the retina or on the image plane. each image Point Q will have coordinates ξ and η.

Planar rotation and movement of the viewpoint allows the user to move directly from the 2D visual image on his or her retina. Visually, you can get a sense of how 4D objects look in 4D space.

This means that the user has already created a 3D unit in normal space from a 2D image of his or her 2D network If This is the same as intuitively sensing a single clit object. Figure 7 shows normal space. It shows the eyes that do not receive help from the one person who is there. This human is 3D like an energized cube View an object from one angle and position. The retina of this human eye is a 3D object Just by looking at it as a 2D fragment of it, we unconsciously but intuitively understand the shape of a cube. Determine where each piece is and form a 3D object there. .

The user can draw a shape on his/her retina based on the 2D image points displayed by the present invention. The same intuition helps when looking at 4D objects from created 2D fragments.

The movement and rotation of the object or its attribution to the user's eyes is superimposed on the user's retina. This is indispensable for organizing the image points.

Before explaining the present invention in detail, we will explain the 3D unified space according to the present invention. D How objects move and rotate 2D- people in a 2D world (i.e. a plane) It would also be useful to know how 3D people view 3D objects in perspective. Just as we see in law, 2D people also see 2D objects in perspective. Similarly 2D The problems people have when viewing 3D images are the same as the problems 3D people have when viewing 4D objects. They have something in common.

In image 8(a), a t2D human sees the 3D world through a transducer. 2D humans have a 3D sense of distance when looking at 2D objects. You will notice that it remains in the space, i.e. the object is perpendicular to the 2D space. As an object moves away from the pupil along a single straight axis, its dimensions decrease in proportion to distance. More specifically, in 2D space, it may appear as a line segment store. The edge of the object being passed over is indicated by the line segment A'B' as it moves away from the viewing axis. The size becomes smaller. Similarly, in 3D space, the line segment ■ of a 2D object is As it moves away on the axis perpendicular to the 2D space, it is shown by the line segment C'D'. The size becomes smaller so that In FIG. 8(b), for simplicity, a one-dimensional retina The eye of W, who has It shows the place. 2 and 熾 in 4D space merge on the user's 20- retina. However, in the same way, here, the y and two axes merge on the one-dimensional retina, and the result is ``progressing straight.'' It looks like "I'm going to go there." Therefore, distinguish the two axes and points p and q on the one-dimensional retina. In order to do this, rotation around the appropriate coordinate axes is required. Figure 8(c) shows the X axis. This shows that rotation around the center is of no use in distinguishing between destruction and path. But the first Figure 8 (D) shows that the rotation around the center can be quickly distinguished from the one axis that takes one axis. , shows that points p and q are distinguished in the same way.

Similarly, when a user looks into a 4D space, a 2D image appears on the user's retina. A point is formed. However, in this case, merging of the V axis and the leakage may occur. You In order to separate the two axes and multiple images that are superimposed on the 111M of the laser, movement and rotation are required. rotation is required. However, although it is L7, the rotation here is not an axial rotation, but around a plane. If the 2D image is separated (i.e. "plane rotation"), the user can distinguish the components of an image of a target object using an intuitive method. next, Further rotation and translation can begin to reassemble these components. This reorganization In this case, the user assembles a 3D object from the 2D components of the image, which is a normal space. It is done in the usual manner. Physically, plane rotation is not possible at the current evolutionary stage. It is possible. However, in the present invention, by using translation and plane rotation, plane motion can be translated into 4D space. It can be attributed to a user viewing a 4D or 3D object in between. This way In this way, we human beings have vision that cannot be obtained with the visual system that we have on the 11th floor of our current evolution. It becomes possible to experience the degree of freedom of effect.

In the present invention, a computer is used as a lance gauge. Thus the compilation The computer serves as a general-purpose optical instrument in creating images of objects in alternating geometry. In contrast, both visual and other means of imaging I will do it.

FIG. 9 depicts the overall functionality of the system of the present invention in the form of a line diagram. ing. The system has an input system for entering the properties of the selected alternating geometry. 10 and one object in the space S1 of alternating geometry. This kind of input system The system may include object position and motion detectors, graphing devices, or a keyboard and controls. The image creation system may include a device for inputting voice commands to the computer. It further includes a transducer 11, which effectively directs the user's eye to the sky in an alternating geometry. 0th order, which determines the direction of the ray of light that reaches the pupil of the eye located there, The transducer is placed in the normal space S where the eye is located. (In other words, 3D Euclidean space ) and each of them intersects a 1uic or drawing plane). A point (ie, an image point) is generated. The system usually emits images of reality throughout space. There is a system 12 that inputs images and presents images according to a selected format. They are Film, tape, computer screen, or stereoscopic screen spectacles, stereophonic sound, shape manipulation handles, etc., and virtual reality. It is designed to generate the sensation of virtual reality.

available to anyone or immediately due to current computer technology Full implementation of the invention on a high-speed computer workstation would make the invention The output of the system can be obtained so quickly that there is no perceivable delay after the input.

Therefore, the system controls the movement of the joystick and the rotation of the trackball for control. Therefore, it immediately reacts and shows the visible image, so the user can see the given alternating space. You can get the pure experience of exercising internally. Within this space, the user You can move freely within a group of objects designated to make up a single world or environment. You can take immediate action and move around. “Graphing/imaging) [ If you do this while in GRAPHING/IMAGINGI mode, the system will For example, it is possible to obtain output from O's image creation system, and It is also possible to graph the four data streams from the system in real time. Ru. Moreover, the user can also change display parameters during the process. child The user can interact with the system to control the process. It is.

Among the main components of the system, the transducer is the most essential part of this new invention. It's a minute. A virtual object defined in the space of alternating geometry by transducer It becomes possible to directly display real images. The functions of this system will be explained first in Section 2. It is described in a more general sense in relation to 4D Euclidean geometry. and the third In this section, we present the simulations for both 4D uniform geometry and 3D Lobatievsky geometry. More specific explanations are provided for each operating mode of the stem. that It should be noted that the present invention can also be applied to alternating geometry. I want to be loved. Here, 4D uniform geometry and 3D Lobatievsky geometry are taken as examples. The above description is not intended to limit the present invention.

Il, Specific Examples A. General functions of the system Referring to Figures 1O and 10(a), alternating geometry, more specifically four-dimensional (4- D) Image processing system and method for objects in Euclidean geometry The procedure in a specific implementation example will be explained. The systems and methods of the present invention are applicable to other embodiments. It can be used in the same way even if the It is provided merely as a means of explanation.

In this embodiment, the input device 14a in FIG. 10(a) is used in the first step of the procedure. to select the system operation mode stored in the mode configuration storage device 14b. Ru. Although many operating modes can exist in a system, the basic mode is has a "visualization and manipulation" mode. This mode includes “go/see/touch” and and ``move/turn/touch'' display modes. These modes are Can be used in conjunction with other modes of the system. Other modes of the system An example of this is the Graph/Image mode, which allows you to Able to draw and display graphs. In addition, it can be used to form geometric shapes. There is also a "blackboard" mode. In this mode, the appropriate CAI required for the purpose) A tool for (compute-paid design) is provided.

In the second step, we will discuss alternating geometry and one or more objects within it. Define the characteristics of This means that the input from input device +4a is predefined. By selecting from the alternating geometry storage device 14c and the object storage device 14d. It is done. A number of system properties define them and describe them in the alternating geometric space S1. It is possible to create, manage, and change the world or environment inside. This is rare 0 essential features that can be performed regardless of the complexity and detail of the desired alternating geometry. One of the possibilities is to choose a certain alternating geometry to obtain a particular mode. this With 0 essential properties that fall into the region of the selected geometry, the process The second is the setting of the coordinate system Σ in alternating geometry. Few further definitions are needed. Definition of an object that is at least one display object of some desired shape will include position and rotation or movement in alternating geometry. other o object is a stationary object, gateway or other object in this alternating geometry. 0 is sometimes defined as a benchmark (reference point) from which to view objects. When the invention is fully implemented to produce a movie or anis, living or non-living Objects that create the shape of living things can be moved at any speed that creates the shape of a living creature. , may perform operations that influence each other. It also moves according to the operator's commands. It can be any type of storage device, but it operates according to instructions from it. It may be as follows.

The next step, the fourth stage, involves moving (shifting) and rotating one or more objects. More will be done. This may be done before displaying the object, or it may be done before displaying the object. Feedback, ergonomics, or interactive input systems after displaying objects Sometimes it is done as part of. Furthermore, location storage device +4e (i.e. position, rotation, etc.) ) may be used to select a pre-programmed continuous display. operator is 2Å or more, use the method described here to connect two or more inputs or steps. applications can be connected to a single system and commands can be issued from them. and prepare for current priorities in system control. Among other characteristics may include colors and animations, for example. Desired level of detail in color and pattern When you select a large number of objects with It can make you angry.

The third stage of the procedure is the final step for the input system. In this stage, input device +4 The eyes of the transducer system are placed in the 4D space S by the use of a. , and is simultaneously defined as a 3D unified space S0 (ie, a normal space). the As a result, a new 3D coordinate system will be set. Thus the transducer The eyes of the system and the corresponding eyes of the user are initially set to 7', :41) Determine the position of the pupil in the standard system, and set it at a position where the direction of the axis of sight can be specified. It will be. All properties of alternating geometry and objects in stage 4 and Eye movements and rotations of objects and/or systems are performed via input device 14a or may vary in various ways as a function of other input devices. properties of the geometry in question Due to the complexity of the properties, there is a wide range of possibilities for using the input system in the geometry of the space S1. A number of decisions may be necessary.

The next 31 to 30 conversion is performed by transducer +1. This is the first 10(a) The CPU 14f in the figure can execute without restrictions. The CPU is the 10th In the fifth stage of the procedure shown in the figure, it actually operates in the two spaces mentioned above. In principle The transducer has a fractional structure that differs from the geometry of the space in which the eye is located. Works when clarity problems arise during geometry. In the present example the invention The general function of the system is explained by Euclidean's four-dimensional geometry for the space S1. be done.

The transducer operates within S1 according to the fractions and theorems of its geometry, and the object of one object in its geometry. In the third stage, the incident rays from each of the vertices points are determined in position and direction at pupil 0. Determine the angle at which the system intersects the eye axis. Next, in the 6th and 7th stages , the transducer moves through the geometrical domain and directs the system's eye rays into the normal space S Determine for equivalent image points within 0. In the 6th stage, the output mode of the system is Selection and space S. performs the scaling necessary to calculate the rays from an object in You have to. An example of a selection mode is wide angle, which shows some kind of optical effect. Examples include using a lens or a telephoto lens. In the seventh stage, the actual image in space S0 is Determine the image point. After that, apart from the transducer, the system's CPU 14f, suitable output means 14g, and output storage device 14h for screen display and Generate the necessary signals for other outputs. In the eighth stage, various Allows control.

For example, some panel controls allow the user to specify the objects specified in the second and third rows. object and/or system eye position (i.e. movement and/or rotation) It is possible to perform various redefinitions of (conversion).

How can a transducer function like this in two mathematically separate worlds? is due to the ability based on the following geometrical circumstances. Please refer to figure 1 for this. I will refer to and explain. In this diagram, 0 is the pupil of the eye, and P is outside the space where the eye is. One point, here assumed as one point in the four-dimensional space S, is the 0-shot In the light, the transducer measures the angle at which the ray PO intersects one axis in the eye system. For example, the X axis in the figure is the point 0, which makes an angle α with PO, and the X coordinate , and P define one plane. This is generally not called a plane in So. However, it is possible to measure the necessary angle α for the time being. of the present invention The transducer thus determines the angles α and β at which the ray OQ intersects the axes X and y. It will be determined according to the laws of appropriate geometry. In the example shown here, determining the angle is quite simple, but in other cases, such as the case of 3I) Lobachevsky geometry. However, things can get complicated.

By passing the corresponding beam to the output system, the transducer is effectively output according to the laws of 3D Euclidean geometry (that is, normal geometry). Determine the direction of the ray. Calculation of two angles, angle α with the X axis and angle β with the y axis, is S0 This is necessary and sufficient for determining the direction of the output ray OQ in space.

In the present invention the eyes of the system are placed directly into the alternating geometry space S.

The line PO drawn from the object point to the pupil of the system's eye is a straight line in that space. It is. In the current example, S is assumed to be a 41) Euclidean space, so 2 This is the Euclidean distance, and the distance r from P to 0 expresses the Pythagorean relationship ( Calculated using However, in this case, the fourth component V is used. Here, system The eye of the mouse is itself a three-dimensional system of ordinary geometry, but it has the features of the present invention. [, on the other hand, the set of all four coordinates of ζ81 is assigned to the pupil. The two and three axes are arranged so that they align with three of the four orthogonal axes of 31. Sometimes it comes. Each coordinate of the pupil is (x,,,”/(1+zkav0)! 2 teichisuera no 1. , the distance from the object to the system's eye is 11 1PO is given by the following formula 6 r = [(X-XQ)2+ (y-yo)2+ (z-z,)2+ (v-v o) 21”’.

Each point of one or more objects is defined in the second stage, but this quantity r must be determined by the transducer in the fifth stage, the result is Stored in column format so that it can be accessed later in the process as needed. be done. This will be explained below.

For each object point P, use the following trigonometric relationships to find the corresponding object point P. The angle α is calculated and stored in the same way.

α=cos”(x/r) Here, X is the X coordinate of P, and r is the value just determined. need one more The angle β is the angle that PO makes with the y-axis, but according to the following relationship corresponding to the above: calculated from

β=cos’(y/r) In the sixth step, the simplest way is to say that the new rays in the space S0 where the eye is located are emitted at the same angle. In such cases, it is sufficient to set α°=α and β゛=β, but , other functions can be used to adjust the effects of various optical lenses. You can do it.

Since the direction of the ray OQ in \ is determined by α° and β′, α′ and β′ Once this is determined, the role of the 1-transducer is completed. Therefore, in the 7th and 9th The role of the output system is to use traditional algebra and trigonometry to Determine the display (e.g. one picture plane on the screen, i.e. the system's eye As shown in Figure 12, this geometry has the following relationship: produce.

r°machi 1/cosγ゛ and CY) Sγ-jl-cOs'α'-eos2β1'/2, Yukote'r゛ is α and β and the distance d to the screen (ie, the focal length).

After determining β′, the output system sets the output coordinates ξ and η in the relationship shown in Figure 13.14. It is decided to set the value to 1 according to the . In this case, η-r'cO5β', The corresponding formula for expressing the plane −C′ on the retina to the X coordinate of a certain ξ is as follows.

ξ − r’cos α.

Interactive visualization and operation mode for operations that change to a specific mode. The system of the present invention uses "line</'see/touchJ" in "interactive visualization and manipulation". QO/1. oOK/TOUCH) is “Do not move/turn/touch J (MOVE/I You can use any view mode of ``tJRN/Y01JCt(). I can do that. In the ``Go 5, Ride. 7' Touch'' mode, the motion is within the alternating geometry. An object is attributed to a user viewing one or more objects. direction of view The viewing axis can be freely selected. Moreover, the user can touch any point within the alternating geometry. It can be moved and/or rotated so that it can be moved. A user - can perform, for example, 4D Euclidean or 3D Lobatievskian motion. have the experience of becoming In this mode, the user/viewer In effect, it is freed from the constraints of normal space. Any object in alternating geometric space or Select a point and perform the necessary virtual movements to see, go to, and You can touch it. These attributed virtual motions are generally less familiar It's the kind of thing that doesn't exist. Rotate without rotating the user's head or limbs around an axis (For convenience of explanation, we use “plane rotation J’plana An example is ``r　rotation'). [move/turn/ In "Touch" mode, one or more selected objects can be moved or rotated. It is visualized and manipulated by doing things like

The embodiments described herein are based on the method and system illustrated in Figures 1O and 10(a). control and viewing of the system is done using a computer, and The design of the control panel and display screen obtained from the computer monitor It can appear. An example of the go/see/touch mode is shown in Figure 15. ing. If fully implemented, what you see on the computer screen here is the computer control that takes the form of an ergonomic reality that is separate and/or remote from the computer. The main part and the reading part. The example shown in Figure 15 is machined using a Matsukin trash computer. Using software called MAT (EMATICA), hyper This can be implemented using a stack (HYPE R3TACK) card. So This is currently being implemented for development purposes because of its convenience and flexibility.

However, HyperCard, which is commonly used in personal computers, Sematism, or other systems, can be invaluable in certain applications. However, the present invention provides an ergonomic and realistic input/output system as intended. This can also be done through more advanced, fast-responsive, and interactive systems. I can do that. The "button" shown on the screen is active and the 15th It can be controlled by touching the screen shown in the figure. Also, "Mouse", "Track" It can also be controlled using a keyboard, keypad, or other device. that accept user commands. The on-screen reader will display the current It shows the coordinates and direction. Panel configuration depends on given geometry and mode of use can be changed to suit. The usage mode is ``Interactive visualization and operation''. ” mode, “blackboard” mode, and “graphing/imaging” mode.

The control panel model in “Interactive Visualization and Manipulation” mode shown in Figure 15. Joules are further illustrated in Figures 16(a)-(d). these modules There are two modes: "Go/See/Touch J mode" and "Move/Turn/Touch" mode. I will explain.

The screen module 20 in FIG. 15 is inside the screen of a computer monitor. window, including any screens, records, printouts, etc. It shows the form of some form of display device.

As shown in FIG. 15 and FIG. 16(a), the Ha sampling module 25 has the current position There is an attitude instruction section 26. “Stay still/turn/touch J In mode, they report the position and orientation of the object being manipulated. Lined up from left to right The instruction unit indicates the X, y, z, and v coordinates of the center of the target object or the reference point. Ru. The remaining two instructions indicate the rotation angle values U and W around the μ plane and W plane, respectively. It shows many things. As shown in Figure 15, in the [Go/See/Touch] mode, the same The same indicator similarly determines the user's position and orientation using an alternating geometry coordinate system. Report.

As shown in FIGS. 15 and 16(b), the moving module 30 has a target object (object). ) (for "move/indent/touch" motes) or Operator (for "go/touch" motes) If you have a convenient type of device to manage the position (in case of “see/touch” mode) Ru. Shown here is an arrangement of several buttons 31 (the same is true for sliders). ) and is designed to control movement on one axis. This motor control Depending on the system's eye-referenced coordinate system or the user's selection, the alternating geometry This is done using absolute coordinates in mathematical space. In this example, the control part of the operation is (or rack ball), active button, or equivalent. I! There are touch screen and position type devices that can be used for a variety of purposes. These are this invention shows an early implementation of the The two buttons 32 at each end are intended to provide control methods for operating the object. interpolate between positions (or double-click the mouse) It is used for the purpose of automatically starting continuous movement to the leading edge of the target side. Shown here In this example, only one axis is under control at a given time. current tube The reading unit 33 shows which axis is under consideration. To select the axis to manage, select You can make immediate changes using the "Select" module.

As shown in FIG. 15 and FIG. 16(c), the rotation module 35 has an active button. has radially arranged displays. Now, from now on, we will use the format shown in Figure 15. This explanation assumes that you will be using the touch screen of the cutter. However, even in this case, the mouse computer, keyboard, and other information input devices. Each button has a Objects (in [Move/Rotate/Touch] mode) rotate around the reference plane. ) or the user (in Go/See/Touch mode) step to each angular position, in this case 6 at a given time. If only one plane is under management, the selected plane is not shown on the display 37. Ru. The button 38 allows continuous movement in 30 degree increments. if mouse Double-click to move the object (or user) in the specified direction. I! It will continue to rotate.

Button 39 is for selecting the plane around which the rotation is to be performed. Ru. Button 39a is the home button, and is used for objects or users. Used when returning the user to the coordinate center or alternating geometry reference position. Button 3 6a is a touch (TOUCH) button. This is the object you see in the output The purpose of this test is to test whether the target is truly “within reach”6. 1. Within reach J refers to a certain specified test interval, with respect to which The details will be explained later in connection with FIG. Operation and correct operation of the invention By this, you can approach any object in the 4D world shown here and "touch" it ('!ou ``ch''), but it seems that it is nearby (because it is an object that can be seen). It's not always within your reach, but when you're able to do it well, The button 36a lights up, or some other force) signal is indicated. .

15 and 16(d), there is a button 41 on the control module 40. As shown in FIG.

This button is pressed when the movement module 30 moves along a certain axis. This is to quickly select the axis. Area 42 is a computer system control buttons (“Go to next screen”, “Go to first screen”, etc.). This is the part that can be placed.

1. [Go/See/Touch J mode in case of 4D Euclidean geometry Next, in detail The explanation is based on the device shown in Figure To(a)) and the hand +1 @ in Figure 10. Go/See/Touch mode for 4D Euclidean geometry This is an example. Specifically, this is a computer shown in Figure 15. It was carried out as shown.

1st stage In the example shown here, [Interaction and Visualization] mode is selected, and a specific model is selected. Select “Go/View/Touch” as the mode. With this mode combination, The user can easily move around in the space S1 of alternating geometry.

2nd stage First, we need to define the properties of alternating geometry. In the current mode, the selected alternating The geometry is 4D Euclidean geometry. We must define the coordinate system Σ. child Here, it is assumed that this consists of orthogonal coordinates (XI'yl' z, , v,).

Other coordinate systems such as logarithmic coordinates, polar coordinates, and other nonlinear coordinate systems are also required. It can be used for The origin of the coordinate system is Ω, and the coordinate system has all zeros as follows. It can be defined as

Ω = (o, o, o, o) Each object 1 point corresponds to the selection of 4 coordinates, which create a vector. Form. The following unit vectors are attached to each of the four coordinate axes: r, 'J+　k, THere, ↑−(1,0,0,0) Any number of objects in the coordinate system Σ of the 4D Euclidean geometry space Sl? It can also be defined in Objects can be streets, buildings, gateways, etc. objects that can be used as benchmarks or benchmarks of any kind. Objects can be viewed in terms of their interrelationships. For example, one object can be interacted with while the other object is used as a background or benchmark. used. Here, every object is expressed as a geometric object in coordinates. It is interpreted as The ta command is also interpreted in addition. First drawn in some form and/or or the stored image with the geometry determined by the vectors defined and described here. The following two items need to be determined here.

(1) the internal configuration of the object, and (2) the position and orientation within the coordinate system.

To define one or more objects, first add a value to the object, such as a vertical angle. It is best to define the decision point by giving it in the coordinate system Σ, then the The object can be moved to any coordinate position (X + Y + z, ■), and the object can be moved to any coordinate position (X + Y + z, By rotating around the reference plane, you can orient it as you want.

An object is defined as a set of vectors in the form:

? i:H-(q,,,・circle,・Qii;chi) Here, the symbol circle is the i-th object This is the ``th'' point. An object like this? −i is one or more points j It will consist of

3rd stage Go/See/Touch 1 option has a sky for the eyes of the transducer system Between S. It is necessary to define the coordinate system of As already explained, the eyes of the system Objects constitute a three-dimensional two-dimensional space. However, it is 4D space S. It is always located and oriented in the coordinate system Σ. At first, the eyes of the system were for convenience. Space S. placed in This is the origin in the normal space from Part 1 ◇(0,0,0) The pupil of the system's eye) coincides with Ω, and its axis (x, y + z) is the axis of Σ (x , +Y1+21). At the beginning, it is a space inside Σ Then, its V is v, -0. Within So, the pupil is at the origin 0, and the retina or The image plane is at a distance d (that is, the focal length) from O, and the coordinates are ξ and η. Ru.

The user is thus represented by the eyes of the system of transducers in Σ.

A transducer converts input information into an output that ultimately Become a sense and reaction. The computer monitor screen in Figures 15 and 16 is the picture plane. act as a ray, producing a ray of light that enters the user's eye.

This is the same as the original ray entering the eye of the system in Σ. ``Paper mask'' The image formed on it is one of the user's eyes! Project a similar image on till It is a plane that can be created. It projects a beam of light into the user's eye and uses the beam to The direct image of an object existing in alternate space appears on the user's retina completely. The final effect is that the user's eyes are shaped as if they were placed directly in the space. have. Therefore, in FIG. 11(a), light from an external point P [PO is a point It forms an angle alpha with the eye X-axis of the transducer system. Simplest implementation , the new ray OQ has the same angle and is placed on the screen or picture plane at Q. The 0 point Q incident on the screen is the ray QS from the 0 point Q, which becomes a light point on the screen. is incident on the user's eye at pupil S, and at the same angle α a point on the retina of the user's eye Reach R. In this way the screen plays the role of an intermediary and the ray QS so that it is incident on the user's eye at the same angle as the screen received. Ray Q R is equal to the ray OQ, and the system uses point 0 effectively as the user's eye, or Practically places the user's eyes in the alternating space so that he or she can directly see the point P. . Since the angles and distances are equal, the screen only plays an intermediary role; In this respect, the present invention has a "cancellation" function as a body process. This is different from the method of drawing images. Other methods of projection such as painting or stereoscopic image projection The purpose is to create a No 17, it doesn't even claim to have that kind of effect.

This can be done by positioning both planes relative to the eye in various ways to vary this effect, e.g. Wide-angle lenses, telephoto lenses, or other options that may result in optical changes. The present invention shown in FIG. 15 does not determine the performance of the invention. j</see/′ In the Touch 17-D, the control panel is the central work input and allows the user to ΣThe aim is to realize virtual reality where you can move as you wish in space. Third The values entered in the rows can be selected by the user in the alternating geometry space. The actual manpower is a parameter governing the location of the destination to be selected, so (joystick, gloves, helmet, etc.).

4th stage In the “row < 27 see/touch” mode, the user can adjust the system's eye position and (-) You can choose the direction of the virtual axis. And user/view two (Kan'sai 8) Control the E-looking J direction, that is, the "forward" direction, and reach there in the "touch" mode. I can do it. Eye position (or object in “move/indent” mode) is interactive I can do things without moving. Changes in the position of the system's eyes correspond to movement and rotation of the system's eyes. This is done based on the line of sight axis. Movement means a change in position, and rotation means a change in position. The effect is generated by influencing the direction of the line axis. In other modes, the Defining the eye position of the stem is not very important, often the eye position is the default Sometimes it is better to leave it as is. A complete execution of an interactive operation comes full circle. It's a cycle of going back. It starts with the input the user makes and the system responds to it. The system will present the corresponding output. Cycles allow users to interact with and create their own Complete by requesting new input that meets the purpose. start the system cycle To do this, the user must move from the original position specified in step 3 to the determined system eye position. It is possible that the movement of In this case, specify the eye position by moving and rotating. actual However, the system's eyes can "go" anywhere in 4D space.

To move, define a new position of the origin ◇ of the pupil of the system's eye in the Σ coordinate of space S1 There is a need to. The definition is given by pect/K>, where ◇−(Xo+YOr ZO+Vo) The new decision point in sigma in 4D space is represented by 0 6 Also, the origin Ω of Σ and the unit The vector is moved with reference to the movement of the system's eye position. new object The origin of the point vector, unit vector, and sigma are moved from the following relationships:

q', -q, -0 1・-↑-6 ni'-k.

↑-1-6 The direction of the viewing axis of the system's eyes can also be changed by plane rotation.

As defined at the beginning in paragraph 3, the line of sight axis of the system's eye remains parallel to itself. In a sense, we are looking at the same point at infinity, the system's eye guide. S Changing the eye orientation of the stem requires rotation in the fourth dimension. This is centered around the axis. It is not a rotation around the center, but a rotation around a plane.

In the current example, several rotations are defined as a set as follows.

Rotation by angle S around the reed plane Rotation by angle t around V plane Rotation of angle U around μ plane Rotation of angle W around W plane The rotations are supported by each corresponding matrix, which is an extension of the familiar rigid rotation in 3D. will be arranged. These will be called T, , T, , Tu, and T. and For example, let's consider the case of rotating by an angle S around the Ashi coordinate plane. The calculation is Q-v+T, "qv 9′ here. is the vector defining the point after translation but before rotation, q− is the rotation Later coordinates, and Ts (matrix indicating the rigid rotation of the angle of fs, i.e., and Ts is the matrix of the rigid rotation of the angle of S Matrix indicating rigid rotation, i.e. cossoo 5ins00 T, - -sins00 ■ss, O oot If the rotation is u degrees instead of S degrees, the rotation matrix would be:

cosu　OO5inu Ol　0　0 u- -sinu　OOcosu The unit vector and the origin Ω of Σ must also be rotated as follows.

i″-Ts @i’.

T--75 and r.

k”−TsΦ′ and 1″-Ts reference I゛.

Ω”−TsΦΩ′ At this point, the motion is attributed to the system's eyes and is brought to the desired position. and the vector q4 of one or more objects, the unit vector, Σ The origin Ω is all S. has been redefined. However, the resulting The orientation of the system must be specified in terms of Σ for the angular position of each of its axes. There is. The resulting S. To determine the orientation of the It is necessary to move the coming unit vector backwards and bring it to the origin. this is By subtracting the coordinates of the origin of the coordinate system Σ from these coordinates, be exposed.

--i"　Ω".

”′″−””　Ω′.

k−−k−Ω” and I-1]″-Ω゛.

The coordinates of the resulting unit vector are its projection onto each of its critical axes. So and these are the direction cosines of the angles that determine the general direction. The line of sight is a system These are the two axes in the eye space S0. Therefore, the line axis is the system's eye axis. These are the two axes in the space S0. Therefore, in order to determine the angle of the line of sight axis with respect to the For this we use the two components of each unit vector and form the following relation from the direction cosine: from which the angles α, β, γ, and δ were determined.

α+Co5tz.

β-CO5''T:'.

Thus the eye directions of the system shown in relation to Σ are the angles α, β, γ, δ and X0+Y (denoted by the 30 position in Σ represented by pZorvo) Ru.

5th stage The new system's eye coordinates, gaze axis direction, and object point coordinates are calculated. Then, the system of the present invention starts the fifth stage conversion function. ray 74, Consider that 5 is radiated from sleep and goes to pupil 6 of the system's eye. Shown in Figure 11 Imagine a plane containing the rays above and the y-axis of the eye system.

Since the circle is generally not within the three-dimensional space S0 of the eye, this space is also not there. but It does not interfere with the use of this transducer.

It is necessary to calculate the angle that this ray makes with the y-axis. For that purpose, first r, It is important to understand that the actual distance from qo. That is to say , perspective effects, the use of stereobuticons, and even sound effects. This is because it shows the effect of the true distance between the operator and the object in the original space. Ru.

Next, the direction cosine of each incident ray is calculated from the following relationship:

λj　””’it　”i μ. =q″, /r.

From the results of the above calculations, we can see that the rays from each object point The angle α between the y-axis and the y-axis. and β. is calculated using the following relationship.

α〜cos−’λ β−ω51λ In this way, for each object point, the angle α. and β. is Qu and 0 exist at the y-axis and the point intersecting the y-axis, respectively.

The fifth stage is completed when these two angles have been calculated for all qij points. Ru. Note that the angle of incidence r, I with respect to the y-axis of the system eye is not used. .

No. 8 Continuous feedback to the first stage enables system implementation at high JI calculation speed. The more you do this, the more rapidly the loop between the eighth and fourth stages can close.

The higher the speed, the more rapidly the loop between the eighth and fourth stages can be closed.

This allows the operator to more immediately feel the reality of movement in four-dimensional space. I can do it.

Step 6.7 Once all the angles of the incident rays for each object point are determined, , the fifth stage is completed and we move on to the input to the l transducer. transju For each object point 1°, image point Qij with coordinates ξ and η is placed on the retina. Or generate it on the screen plane. A transducer points to each object. to the angle of incidence α, and β. Determine a new ray in the system's eye space from and make it visible to the user. In the sixth step, the selection of modes of presentation and scaling is necessary. The ray corresponding to the object point is the eye knowledge of the system in space S0. You can adjust the sense as you wish and project it in the desired mode. This includes the output light By changing the line determination, you can create the effect of viewing from a telephoto or wide-angle lens, or create an arbitrary Methods such as using optical lenses are used. How to use the output of the system is in step 7 In this example, the output is for use in the final presentation in the form of a computer or screen; and the selection in step 6 is scale setting, so the new The new rays are projected at the same angle in the system's eye space S0. Therefore, here Let α, and β be the projection angles in the eye space of the system, and α′ − α. and β′ It is sufficient to set Yuichi βU. However, you can create other modes and optical effects if you wish. If you do not take into account the optical effects of wide-angle or telephoto lenses, From the homogeneity of the function, α', -r(αg)I, where "(α5) It is necessary to adjust the function l taken into consideration.

In the simplest case pursued here, the system output is the computer monitor. The image is displayed cleanly and presented to the human eye. As mentioned earlier, this It represents the retina of the human eye. One image on the retina of the user's eye (that is on the screen) ), the system's eye focal length d0 is set by the user Normally, the unit should be set so that the reading distance at which the monitor is viewed is equal to the reading distance. In a world with users, this is about X inches (about 51 centimeters). convenience Let this be the natural unit length in the 4D space of the current example.

In the next seventh stage, determine the ray in the space S0 where the object point is located, and An image point Q, is formed at the retina or image plane L of the system's eye. first new occurrence The angle γ' that each of the rays made with respect to the two axes in the milk is determined. here The angle 7・' on the y-axis of 30 It is important to remember that the calculation is based on logic. As in the current example, the corner Even if we consider that degrees α” and β′ are not equal to α and β, respectively, γ゛, This is not the case, and γ°− is calculated from the following relational expression.

γ',=cos'[1-λ2. −u 2j+''Next, a certain object point Determine the distance at 30 for the ray. The distance is represented by R. The relationship is as follows It is as below.

R,=dSγ'.

After fi, determine the image point on the retina or image plane of the system's eye in another space. The 0 image point represents the corresponding object point that has been transformed one by one, and is expressed by the coordinates ξ and η. be done. The coordinates of each image point are determined from the following relational expression.

ξ,=R1 cos α′i η. −R, cosβ′,.

8th stage At this stage, output control for display and further interactive manipulation of the displayed objects are required. Set up the control section to do this. Any given device has an image point (ξ., η,) It is possible to have any number of alternate modes for assembly and use. shown here In the example “visualize and manipulate” mode, the output is sent to the computer monitor screen. It will be done. The assembled and transformed image points are then delivered to the ninth output stage. It will be done.

In the explanation of the [Go/View/Touch] mode, we will explain the controls on the computer panel first. Settings in the control section were like adjusting the display for the user.

That is, it looks at one or more objects that can be adjusted at a moment's notice. In fact, in that mode, the display Any changes that occur are also sent back to the fourth stage, where they are updated with a completely new set of outputs. is starting to occur. In this sense, this system can be said to be completely interactive. I can do it.

“Touch” or [TOUCH] command in the 8th stage (16th (C) According to the control of button 36a) in the figure, the user can A point in an alternating geometry such as a point can actually touch an object. object The object is located in the same space s0, and the user takes a step toward the object. touch an object only if it is within a specified reaching distance where you can touch the object. You can touch it. Touch for objects in 4D space on the viewing axis Two criteria are used to apply the test.

(1) The distance from object 1 to the point or point to be touched is set in advance. (2) the distance of each object must be “reachable” within the specified maximum value; The fourth coordinate or V coordinate of the object point is zero and the object point is be in the subject's eye space S0 and not be perpendicular to it at a distance.

If these requirements are met, the object can be touched using touch commands. Rukoto can.

A more detailed explanation of applying this test to the current example is shown in the flowchart in Figure 17. As shown in the figure below. Figure 17 shows the 0th order test with touch test selected. The object point nail is selected. The touch criterion is selected and the hand reaches out to touch the object. The maximum test distance (d,) for touching the object 1- is defined as ≦2d, ing. Moreover, a point or points exist in the eye space of the system. The criteria must be entered and the respective test requirements verified. Ru. If qv(7)v coordinate is 0 and r, is equal to or smaller than 2d0. Then, the object is S. The condition is that it is within the area and within touching distance. The matter has been confirmed. However, if any of these conditions are not met, If it is, the result is negative and the object cannot be touched, in which case The control panel on the 8th stage operates the system's eye and its line of sight axis. shall ensure that the requirements of the

9th stage ``A processing device such as a computer as described above is used in the eighth stage in connection with the present invention. Using this, image points Quth can be assembled. The assembled item is in the 9th stage. and, in the example I will now explain, it will be output to the monitor. The assembled image points are 4D unique The output information represents one or more objects appearing in the code space S1. Can be configured to fit any display device, such as videotape, film, etc. Or can be used with other existing recording devices or computerized drawings It can also be input into the system and CAD system. Other movies and videotape animations computers in various forms, including but not limited to It can be used in the form of data processing. When used with these display devices, the transducer The eyes of the server's system represent the user. Transducer outputs input information It converts into information and ultimately reaches the user's perception and reaction. The output is converted into an image. It can be printed as a picture. The original ray reaches the eye of the system in alternating geometry space S1. In this example, the computer monitor is the output device, and it acts as the screen plane. The rays displayed on the screen are designed to receive the rays of light and display them. That's why it goes into . As explained in paragraph 6, the user controls the scale. deer However, in normal mode, the angle of incidence at the system's eye and the user's eye are equal. subordinate Therefore, the system ultimately displays what the system's eyes itself see on the retina. This will be delivered to the operator's eyes as an image. In this sense, the present invention The entire system is a computerized optical instrument that extracts objects from the space of alternating geometry. rays of light that ultimately present them as images on the retina of the human eye.

touch Ia, visual example of “go/see/touch” mode in 4D Euclidean geometry For the visual example below, the device mode (first stage) is ``go/see/touch''. ” and chose 4I) Euclidean geometry as alternating geometry.

It is necessary to define a set of objects in space S1 (second stage) 6 objects The ect can be in 3D, but it must be in the space of 4D Euclidean geometry. . The generation of light rays in the 3D unified space, which has already been explained in steps 1 to 9, is A system of transducers used to determine what appears on the retina of the eye. used. Something like a gateway is set at the origin Ω of the coordinate system Σ in the four-dimensional space S1. be done. In this example, the gateway is called the arch at the origin. then two Set a benchmark and mark the course in two directions of coordinates. one is far away One marker is along two axes (the line-of-sight axis), the other is at an equal distance and is along the v-axis. It extends into the fourth dimension. These two markers are "Z operation risk" and "■ It's called "operisk". The distance from the pupil to the retina of the system's eye [do is a kind of auto Initially, these two operational risks were placed on their respective axes. is at a distance of too many units from the origin along the line. If the system's eye retina is there The image is delivered to the human eye at the same angle and at a normal viewing distance. Then, the natural unit for d is approximately X inches (approximately 51 centimeters). This is the observation distance. And the operisk is about one-third of a mile from the arch (approximately 536 meters) away. In this virtual world, the basis of these third orders is Σ Arrive at the next coordinate in the standard system.

Arch at origin: (o, o, o, o) Z operation risk: (0, 0, +000 ．．　O)■Operisk:　(0,O,0,1000).

The choice of where to view the object is arbitrary. I'm trying to get a little perspective For example, if you place the user 300 units away along the z-axis and Make it high. The user's eye, which is represented by the system's eye, has coordinates (0, 10, - 300,0>, as shown in Figure 18(a), the object in the fourth dimension is The location of the kuto extends trigonometry and the Pythagorean theorem to include the fourth coordinate in the fourth dimension. Decide using what you learned.

Beyond the arch and the origin of the coordinate system Σ in the space S1, the user sees the first image in the fourth dimension. You can see ■Operisk as a statue. Z-operisk is a little far behind it. It's there. Due to the overlap of images on the user's retina, the user 6. The light rays from the operational risk do not have an X component, so To represent them, the image points generated by the transducer are aligned directly along the viewing axis. become. The line-of-sight axis serves as the z-axis and the v-axis, and from the two axes images are superimposed on the retina. Users are complicated due to packing 4 dimensions into 2 dimensions. I need to remember to keep things tidy.

By rotating the viewing axis as little as possible, the two axes can be separated.

FIG. 18(b) shows the result when rotated by 4 degrees. However, this is normal It's not a rotation in the sense of the word. Users are accustomed to rotating around an axis, but I'm not familiar with the rotation around the yv plane, here the rotation is around the yv plane. be exposed. In other words, the point on the plane whose x and y coordinates are missing) is exactly It is rigid and unmoving as the axis in case of 3D rotation, while other points are 3 Rotate the rigidity in the same way as for dimensions.

Now, the ■axis clearly represents the 0th dimension, and the user can perform virtual operations within the fourth dimension. can perform movements. In this example, the user enters directly into the fourth dimension and selects the ■ axis. You will go down and actually touch the V-operisk.

The user uses the control unit described in step 8 to move down the v-axis to a certain point. Book The inventive system allows the eyes of the system to be placed in a location selected by the user.

In this example, the location is a certain distance from the operational risk and far enough away to see the scene. I chose the right location. After entering 900 units into the fourth dimension, ■From the operational risk are 100 units apart. The user then sets X and the second dimension to zero, and the system's eye Leave as many y values as necessary to place (effectively the user's eyes) at approximately eye level. This movement is shown in FIG. 13(c).

In Figure 18(c), the oprisk is standing in front of the user. The user is now I even looked at the entrance and door handles attached to the operation risk. the user moves forward I felt like I was being turned into a door handle, and I was so confused by what I saw that I gradually got used to it. The arch at the origin can be seen quite far away (900 units towards the background). Ru. On the other hand, the previously visible z-axis now points toward the operational risk that was not removed in this example. It changed direction by only 4 degrees.

The user then takes a step forward and attempts to become the door handle. In this example, 25th rank If so, the distance is sufficiently close, and the key point of the touch test l- as shown in Fig. 18(d). It is possible to meet the requirements. The user moves the system's eyes as much as necessary to In order to approach the position and touch the door handle, the user must perform the following two tests: Must take a bus.

(1) The user must be within the testing distance from the operational risk.

It is set to 25 early 6 and (2) the user has an object to touch. must be in the same space as the object, so the object must be in the same space as the system must exist in the eye space S0.

In this case, the calculated distance to the door handle is 20.0432 units. this This is because the user's eye through the system's eye was higher than the door handle. , which is sufficiently close to the test limit of 25, so the first criterion is met.

The system then tests whether the door handle is within the user's reach or not. This is the object relative to the system's eye coordinate system. is determined by generating a vector that measures the relationship between here The vector is (0,8321,-1,01,0,20).

The first two coordinates (ie coordinates in 3D space) are understandable. That is, it is a little It is on the right side (not on the center line). and a little lower than eye level It has become. However, the fact that it contains a 20 unit α component indicates that it is truly not on the user's line of sight (which is supposed to be the way) and still It shows that it is far away (that is, it is still outside the spatial equation and in another dimension). Yu The light beam that the user sees is created by the transducer. and The entrance door is actually an illusion.

The problem in this case is that the user is still effectively in a space perpendicular to the V-axis. Ru. The user is standing at a point on the fourth axis but is still facing the object. The user remains oriented not orthogonally to the axis on which he stands . If the user takes a step forward and is about to be grabbed by the door handle, the user The person ends up walking in a direction perpendicular to that direction. solve this problem To do this, the user must rotate around one plane, so Then, the 2nd axis (the user's original forward direction) is changed to the V-axis (the new forward direction towards the operational risk). ” direction). User replaces old Z axis with new V Therefore, X and y must remain unchanged. And the rotation is around the q plane must be done. As mentioned above, this is Section 1II A, 1. -9 Generated using the technique described in vgL.

The fourth dimensional view into which the user has entered is shown in Figure 18(d). This is Yu. It was the same scene that =Su'- saw before she tried to take a step forward.

The 4th dimension was replaced by the 3rd dimension by Daqing's plane rotation, but what does this mean? It looks like things haven't changed. However, I have previously seen illusions of the fourth dimension. However, by taking a step forward, the user was pulled back to the third dimension, Now we are beginning to see the Opelsk in its true position. The arch and the end are now illusions ing. On the other hand, the operating risk, door, and door handle are all in the same space. .

Forward movement causes the user to move directly on the fourth axis, and the user You can reach the door and touch the handle of the Opelsk door. This is certain You idiot, take a look at the vector on the door handle. If the user previously 0.8231. -1.01.0.20) and now the user is a vector Find (0,8231,-1,01,20,O). The third and fourth components are the main They actually swapped positions. lay between the user and the door along the axis The distance in X units is still in the fourth dimension, perpendicular to the user's line of sight axis and forward direction. I rotated the system's eyes to the plane so the object is now in front of the user. be.

The door handle is an object in the Σ coordinate system of the four-dimensional space S1. In Sl' No place is closed to users. A single image on the retina is The user's overall standard of spatial perception differs, so the user uses these compressed Effective perception is reached by collectively handling a series of 2D images, resulting in a standard Go to the desired location through regular steps. It enters the space of the object. As the user gains experience with the invention, The user performs large plane rotations first, then steadily and confidently. So, you will learn how to enter the fourth dimension.

2. “Go/See/Touch J mode” in 3D Lobachi Ski Geometric When Lobachevsky (hyperbolic) geometry is selected in “Touch/Touch” mode, The flowchart in FIG. 10 is also the same. However, the generation of light rays in the fifth stage is The details vary considerably. Again, the object interacts with the user's commands. Defined as a vector by discourse operations. However, the object vector It is much more complicated to move from angle of incidence α to @O on the system's eye to βμ It's going to be a great job.

In the second stage, the alternating geometry and the properties of one or more objects are expressed in hyperbolic coordinates. Differences in transducers that must be defined in Δ This is seen in the fifth stage, which determines the angle of incidence of the ray from the target. Here, the line and angle are 3 D must be determined in the alternation space SI of Lobatievsky geometry. There The operations must be performed according to the rules of hyperbolic trigonometry. (Hyperbolic Regarding trigonometry, see Wolfe, for example.

H, E, Introduction to Non-Euclidean (See Chapter 5 of Geome+ (New York 1954)) A decision must first be made about the coordinate system to be used in Chevsky geometry. For the example shown here, it is convenient to select the Lobachevsky coordinate system. be. The point P shown in this coordinate system is shown in Figure 19(a) (Wolfe, p, + 38). What is important here is that in Fig. 19(a) the distance iAP is It is to know that it is not equal to the distance MoB that we take as the X coordinate of P.

In determining the angle at which the eyes of the transducer system effectively view point P, , we have to determine a certain hyperbolic right triangle. The pupil of the eye is No. 19(b)-1 9(c) If it is located at a distance d as shown in figure 9(c), then the hyperbolic triangle of the right triangle The rules of law give the following relation:

-nλ-tanh x / 5inh d.

eosh s-cosh d cosh x.

Shima θ-tanh y/5inh s.

Using these three relational expressions, the two angles λ and θ can be determined, and the angles λ and θ can be determined at point O. The angle at which the light beam enters the pupil of the transducer's eye can be measured.

These angles are not the same as those determined for 4D Euclidean geometry. Be careful. Therefore, reduction to coordinates on the retina or on the picture plane (the first Stages 5-7 in Figure 1O) will also change.

It is also true that this is now a trigonometry problem ``within'' Euclidean space with eyes. Mind you, this happened because the transducer did its job. did Therefore, we will follow the familiar format already explained in paragraphs 6-7 shown in Figure 10. That will happen. Here, ξ and η are 111 points or orthogonal coordinates of the image plane. if IP from the eyes! If we use the distance to the membrane as a unit element for convenience, the relationship in Figure 19(d) is It is as follows.

ξ-tan^ η-1anθ/cos^.

3. In 4D Euclidean geometry [7 times of movement / J mode of touching [7 times of movement] A transducer in the [go/touch] mode described in Section 111A, 1 It is essentially the same as a "see/touch" mode transducer. operation The sequence of operations is also the same as that shown in Figure 10. The main difference is this mode. The system's eye is fixed, and the object is relative to the system's eye. This means that it is operated by Therefore, in the second stage, the coordinate system Σ of the space S1 , in addition to defining the unit vector r+　L　L　r* and the origin bean, the 4D Yuk Define a lid coordinate system R6 for object 1- and find its corresponding unit vector. The origin of Ro with the coordinates (o, o, o, o) Point stations also need to be defined. In the second stage, the user selects the object to be manipulated. redefine the object. This includes the components of the object in the form of a vector, the vector p − is defined as Objects used as other benchmarks or backgrounds The vector i is defined as before by the vector i. In the fourth stage, movement in this mode is This is done with respect to the object and its coordinate system R8. In this way, this mode In the fourth step, the position of the origin in the coordinate system Σ is set as '5;-(x, y, z, v-) It also includes the work of finding it. Then, the object pu with the unit vector is It is transformed into the coordinate system Σ using the following relationship.

f + f + G Official=1+◇ ■・－Inn layer In the third stage, the system's eye position is set to ``Go/See/Touch J mode as shown above. There are also differences in the rotation of the first and fourth stages, which must be defined as follows. That's what I mean is now rotated under the object, not in the eye of the system. Ru. First, under the object. The coordinate system R of is oriented in the coordinate system Σ of the alternating space S1. It will be done. The specified plane rotations of s, t, u, and w are the same for zero-order objects. One of the plane rotations must be selected to rotate the vector 〒,I. for example , if the plane rotation is a rotation of degrees around 5, then the representative rotation matrix is cosk O5ink 0 Q- -sink Ocosk O o 0 0 0 Object hect l-p. rotations and their corresponding unit vectors εj,i, T+ is then determined by the following relationship.

Lower pa. -I'Q"l"ii 'F' = To' 2' ト”−Te↑ Yi・-T、Suke・ Similarly, with the rotation of the other background object, the origin Ω′ of Σ, and the coordinate system The unit vector of Σ must also be determined. As a result, 5-4 mutually-1, Ω-2 and 1-2?・The size of the legs is determined individually.

Using the resulting vector, the object vector p, ts.

It is necessary to decide in relation to The following relational expression is used for this.

Lower U′″″ town +5 and “ Next, Pect l C1η BiQ,,,s-,,Ba,1ゝ,? −，? ’, i”, 1 -, the remaining 6th to 9th stages are ``Go/See/Touch J mode''. As previously described in Section 1II, A.l, and carried out as known to those skilled in the relevant art. It will be done.

4. “No moving 7 times/touching” mode in 3D Robotievsky geometry already explained As mentioned above, the selection of Robochewski (hyperbolic) geometry simply requires the flow of Figure 1O. - alternating geometry and defining properties of one or more objects in a chart The only difference is the trigonometric relationship used in . The difference is in the transducer. It happens inside. Specifically, the incident angle of the ray from the object in the fifth stage There are differences in the occurrence of This mode converts lines and angles into 3D Lobatievsky geometry. The operation must be performed in the alternating space S1, where the operation is performed according to the rules of hyperbolic trigonometry. However, these operations are well known and , their addition is simply an alternative trigonometry exercise for those skilled in the art. It is simply a matter of using the person in charge. However, they are specified in Section 3 A, 2 above [ Lobachevki's explanation of the "Go/See/Touch" mode - a little more detailed in the 3D geometry section It has been described extensively. 41) “Go/See/Touch” in the “Move/Turn/Touch” mode. The outline in Section 1II A, 3 regarding the differences from the “mode that is used” is the same here. As this applies, there is no need to discuss it in further detail.

8. Graphing/image creation mode in 4D unified geometric characters Graphing/image Creation mode allows Needy to graph functions in alternating geometry. Ru. Therefore, this is a measurement instrument, an image forming device, or a data source. Data input can be graphed and visualized, and variables in alternating geometry can be It is useful for examining the relationship between In Graph mode, objects are This is a graph in substitute geometry. It is, as already mentioned, "Interactive visualization and editing" Machining J mode 1 Go/See/Touch and Move/Turn/Touch modes In the process, it is possible to visualize and edit using the tracker of the present invention. come. In this way, each point or object in the alternating geometry is shown in Figure 1O. According to the flowchart, the corresponding image points can be converted. vision The above intuition has been of great help in understanding the relationships between interrelated quantities since Descartes' time. The "graphing/image creation" mode in the present invention is known to be This advantage can be used, for example, when a set of four related measurements is An image-making device that does not only view objects from a three-dimensional perspective? The present invention can also be used for data such as four-dimensional graphing and image creation. Whether you are an engineer, scientist, medical professional, artist, or dataset Available at any time for those involved in the refinement of information and the visualization of relationships in space. This is provided as a method.

1. Hypergraph Instructor In order to explain the system and method of graphing/imaging mode, we Let's take the case of lid geometry as an example. The first problem is in a four-dimensional space. The purpose is to explain how to graph a single point. As a rule, one point at a time is Once plotted, you can plot any number of points in that pond, so the plot is complete. You can create images or graphs in the same way. Generally, the steps 10(a) and the apparatus is the same as that already described in connection with FIG. 10(a). It's the same as what you did.

Figures 19(a) to 19(b) show "high Figures 19(a) to 19(h) showing "Pergraph Instructor" are pine It is a continuous diagram created as a hypercard running on the Kintrash computer. Ru.

However, the same effect can be achieved with other programming techniques, especially with much faster interactions. 0 creation that can be run on other computers, including type workstations. The resulting diagram is computer-generated and uses a transducer. This can only be achieved with a system that uses

In this example, we will use the Matsukin Trash computer, and use the Mathematics power as software. Realized using.

In conventional graphing, each point is the coordinate of that point obtained by projecting it onto the coordinate axes. The position is given by. To plot a single point, coordinate values are given on the axis. It will be done. A projection is made from the axis, and at a coincident singular intersection point it is projected onto a plane or into space. The position of the point is given. Which 4D choice is “Hypergerapin Structor”? Do this also for the points marked. Panel control can be in any desired mode An example of refinement is shown in FIG. These settings By changing the resulting feature, you can view it from any position and from any angle. You can also view it from anywhere. Therefore, this makes the illumination of the object more You can get a general feel. Figure 19 shows a four-dimensional Euclidean geometry. In the plot of point P in mathematics, at coordinates (2, 6, 4, 2), the μ coordinate plane rotation is The 0 point (2,6), which is at 45 degrees (i.e. U''), first gives its position on the x, y plane. It will be done. Then the point (4,2) is given a position on the z,v plane. The point itself is As shown in Figure 19(C), the intersection of planes that cross these points and are parallel to the axial plane It is. The transducer of the invention finds that intersection and presents it to the eye.

The hypergelafin constructor graphs a 4D point P from its component coordinates.

Conversely, component coordinates can also be determined from points in 4D space.

The system of the invention makes it possible to graph pairs of numbers as a function of the pair of numbers. . In other words, a plane can be mapped to itself as a surface of a 4D unified space. It can be imaged. A surface or graph is like any other 4D object ``Go/See/Touch'' in the ``Interactive Visualization and Editing'' mode explained earlier. Examined by movement and plane rotation using the mode and "move/rotate/touch" mode You can.

Figures 20(b)-20(j) show the points (2, 6, 4) of the point P shown in Figure 20(a). , 2) illustrates graphing. First, the user finds a point like Q somewhere on the plane. As shown in Figure 20(b), the point Q has the following coordinates. One.

x-2,y-6゜ Since the point Q is on the plane, its 2 and V coordinates are 0.The user selects the 20th ( c) Determine point Q by drawing a plane parallel to the V plane across point Q as shown in the figure. Project into the 4D graph space. All points on this plane have values of 2 and V and share q coordinates 2 and 6. "Hypergraph Instructor" was explained earlier This "drawing" is possible using a transducer method. The result is This is shown in FIG. 20(d). In Figure 20(d), the plane drawn by the user Note that the surface is not in 3D space, described by the (x, y, z) coordinate system. Well, the value of ■ at all points in this coordinate system is v-0.

Generally, in 4D, 2,000 planes intersect at one point. Here, the coordinate plane and new plane are at point Q. They only meet when they are together.

Next, the user takes an arbitrary second point R on the μ plane as shown in FIG. 20(e). . here. R happens to have the following coordinates.

z　−4、v　−2゜ The purpose is to plot a point P whose coordinates are a combination of Q and R in 4D space. be. That is, X-2, y-6, zhan4. v-2 The user then draws a plane across R to project R into 4D space instead. . 7 This time, as shown in Figure 20(f), this new plane is drawn parallel to the α plane. In the plane, the Z and V coordinates remain constant. On the other hand, the X and y coordinates can be changed freely within a certain range. do.

In Rt&, the user makes the two projection planes intersect. Then, as shown in Fig. 20(a), A point P (2, 6, 4, 2) is obtained in the 4D unified space as shown below. General Since the two planes exist in distinct 3D space, they can be However, the use of the invention , the user looks into the 4D unified space and finds that the two planes intersect at point P. The 0 point P that the user notices is the one that the user can see with the help of Hypergelafin Structor. That's what I was thinking of adding.

The whole process can also proceed in the opposite direction. That is, 4 shown in FIG. 20(a) Starting from point P in D space, its component coordinates can be set to 0. For example, point P? However, in both cases, the given point P It passes through the plane. First, the user projects “forward” from P onto the α plane. Find the intersection Q of the projection plane and the plane as shown in Figure 20(g). As mentioned above, the projection plane and the α plane belong to a clear 3D space, so the two planes are The 0-order user, which intersects only at Q, projects downward from P to the V plane. do. As shown in Figure 20(h), the only point where the projection plane intersects the α plane is R It is.

In this way, the user is given four The component of P, which is the point D, is left. Graph construction in the 4D unified space shown above The result of the present invention is to transform a graph in an alternating geometric space so that it can be seen by the user's eyes. It is a fruit. The graphing mode is the “go/see/touch” mode explained earlier and the “movement” mode. The images are visualized as shown in the figure using the "Scatter/Concavity/Euclidean" mode. child Each object point or graph point represented in 4D space is This is done according to the flowchart in Figure 10 and the selected geometry of the 4D Euclidean geometry. According to the two-mode example in the math, the system's eye changes to the corresponding image point in the spatial milk. will be replaced.

Complex functions and variables for λ graphing/image creation The apparatus shown in Figure 10(a) is shown in Figure 1O. The system used in the procedure also visualizes grid points of Cartesian coordinates in 4D space, and As an example of 0 that can be plotted in it as a representative form of various functions including Figure 21(a) shows the Cartesian coordinate plane in four dimensions. The plane shown in the figure is a Cartesian coordinate system in 4D space using the system and method of the present invention. This is what was visualized. In Figures 21(c)-(d) this plane is the usual seat. Shown with standard system. A point on a plane is effectively a 4D orthographic projection method. projected into a coordinate system. If three planes intersect at the origin (which is true) ) shapes, the various shapes overlap and become visually confusing. There These shapes are shown in an expanded form of the conventional 3D orthographic projection method. This expansion module The codes form a systematic orthographic projection in 4D space. Figure 21(a) shows the standard and The center of these axes is at the origin of FIG. 21(b). No. The projection onto the q-plane in Figure 21(a) forms a customary lattice on that plane.

On the other hand, in the projection onto the V plane, the z and y axes are perpendicular to the A cult lattice is formed. placed in the usual way on the coordinates x, y in the first coordinate frame. Any point P projected onto point S in the 4D system.0 Point S then projected onto point R on the W plane. . In this way, all points of any locus selected on the E plane are imaged on the W plane. It will be done.

The present invention solves the problem of complex graphing functions. In the example below, Commonly used notation is used. One complex number is one number pair, and the real part and the imaginary part. Therefore, one function wr(z) is written, then W, which is one pair of numbers, is a function of 2, which is another pair of numbers. do. In this way four quantities are related. 2 components are defined as x, y be done. The components of W are defined as u and v. Therefore, in the following explanation and attached figures, In accordance with convention, variables and coordinate axes are expressed as (X+Y+u,■>6 However, please note that this method is different from the notation used in other parts of this application. I'll keep it. To explain such relationships within the limits of three-dimensional space, we often use conformal mapping. A statue is used. This traditional method draws two shapes. One is on one side. This is a locus representing the selection of LY values of 2 and 2. The other is in the W plane, and the first It emerges from the shape by the operation of its function. The shapes in the two planes are mapped onto the W plane. That's what I say.

Functions are considered in terms of two types of relationships. Visualizing this mapping in four dimensions would not be possible without the present invention.

The transducer allows for the creation of a four-dimensional graph directly representative of the function, Z and W It is possible to present a locus in a plane as a projection from this one graph.

This graph presents the entire function to the system's eye and to the human eye.

Figures 22(a)-22(d) illustrate the system and method of the present invention in a hypergroup. This is an example specially made using rough mode. In this mode, this relationship is often One function that is often considered is graphed in four dimensions. The function is a hyperbolic cosine, It is written in complex number notation as follows.

w　-cosh　(z) As this hyperbolic function shows, a straight line parallel to the y-axis in the tsubo plane is an ellipse in the W plane. mapped. It is difficult to imagine the form of a four-dimensional graph that achieves this kind of change. This example is shown here for the sake of illustration.

What is shown on the right side in Figure 22(a) is a collection of lines in the (x, y) plane. , these lines are parallel to the y-axis. Therefore, these lines represent continuous constant values of y. represent. These lines are traced onto the 4D surface on the left and this table is used as a parameter. Copy the surface as seen from the observation point y. This table is shown below the 4D surface. It is the projection of the surface onto the (u,v) plane. This is similarly copied from continuous values of y. taken. This procedure of projecting onto the (x, y) and (u, v) planes from Figure 22(b) is equivalent to the usual 3D orthographic projection, and therefore can be called a 4D orthographic projection. I can do it.

The same procedure is followed in FIG. 22(b). However, in this case (x, y ) plane line is parallel to the y-axis and represents a continuous constant value of The 4D surface shown on the side is copied as seen from the observation point of It is given as a second trace of shape. Below the 4D surface in Figure 22(b) are (u,v) A projection onto a plane is made.

In Figure 22(c), the two projections in Figures 22(a) and 22(b) are combined. A single visual impression of the 4D surface shape is made. This is a complex hyperbolic cosine function It is a 4D graph. In FIG. 22(d), four coordinate axes are shown.

Their composition changes as a result of the selection of observation points from which these shapes are viewed. here In , the user's eyes are moved from the origin and the viewing axis is tilted.

As a result, visually these axes are separated and a coordinate plane is visualized.

Also, a rotation of 30 degrees around the (y, z) plane is given to the graph itself. There is.

Similarly, in Figures 23(a)-23(d), the group of 4-dimensional cosh (z) is Show the rough and trace the projection. As a result, a certain straight line in two planes is Map an ellipse to the plane of the attribute variables using a space curve. Therefore, the 23rd (a) In the figure, a certain negative value of being represented. In Figure 23(b), the same line segment is a space curve a in the 4D graph. Project 'b'. This line segment is then transformed into a (u, v) plane as shown in Figure 23(c). Project lo onto the surface (7: ellipse line, ll). In this way, the (x,y) plane The straight line is transformed by the space curve to become a curve on the (u, v) plane. It becomes possible to intuit. In Figure 23(d), the coordinate axes are the reference as before. How is it shown?

The graphs in Figures 22(a)-22(d) and 23(a)-23(d) are simple. This is shown in the form of a wire frame. A full implementation of hypergraph mode The invention was the creation of shadows and illumination (this is a 4-dimensional light that is placed at a position chosen by the viewer). The graph is easier to read because it includes the technique of . However, these data are complex and require new methods to be used intuitively. It takes time and practice to develop the intuition necessary to This object is to the human soul. There are many places that are like ``unexplored regions.'' When you develop intuition of 0 or higher, you can become 4D. Imaging in many sciences, technologies, and other fields such as economics and social sciences uses data. It becomes a daily occurrence in the field in which it is used.

Spatial intuition is required to interpret the extreme complexity shown in Figures 23(a)-23(d). As an aid to the In Figures 24(a)-24(d), the ability to manipulate shapes is useful. The plane is centered around the (y, z) plane in the J-mode manner described earlier. Then, it is continuously rotated by ω degrees, 120 degrees, 150 degrees, and 180 degrees. which one even if they are viewed from the user's position as shown in Figures 22 and 23. ing. Looking at the case of ω degrees shown in Figure 24(a), in this orientation, (x, y) The plane is viewed from the side, while the (u,v) plane is fully extended. Second 4(a) The coordinate form shown as a reference in the lower right of the figure is such that the state forward axis and the y coordinate axis overlap. This is to confirm that it is visible. In contrast, in Figure 24(c) In the 150 degree case shown, the (x, y) plane is fully extended, while the (u, v) The plane appears compressed.

The shape of the graph is as shown in Figure 24(b) for 120 degree rotation, or as shown in Figure 24(d). In the case of 180 degrees, the shape looks very different6. By looking at continuous rotations, and also by looking at other planes of rotation, we can develop intuitive visual concepts of the form of mathematical functions. You can get what you say.

The spatial shape representing the cosh (z) function is ω degrees, 120 degrees, and 15 degrees in 4D space. When rotated at 0 and 180 degrees, features that are otherwise confusing can be distinguished. As a result, you will be able to intuit the form that is synthesized from several views.

Regarding this, a similar situation is that we never [see] the entire 3D cube. Although I have never seen a lot of partial angles, I have assembled my impressions based on my experience of looking at many parts of the plane. It is helpful to remember the intuition and formation of a square, Sections 24(a)-24(d) ) In addition to rotating the image as shown in the figure, you can also use [Move/Rotate] to change the observation position in J mode. You can also create other types of views by changing it.

Function intuition can also be strengthened by ``partitioning'' it in other ways. come. In Figures 25(a) to 25(e), the shape in the same space is not in the W plane, ■It is projected onto a flat surface. This consists of the actual components of the dependent variable and independent variable, so We can see that the more familiar relationship u-cosh x is included in the complex shape. Karu. In Figure 25(a) a new segmentation plane is shown, on which the shape is projected. It is supposed to be shadowed. In Figures 25(b) and 25(c), (u, v) The traces of two sets of curves in the plane are shown. These two sets of curves are In Figure 25(d), they are combined in the (u, v) plane to form a complete shape. It is shown. This is in the form of a hanging cable and is related to catenary lines. can be recognized as such. As shown in Figure 25(e) at the bottom, change the coordinates and enlarge. As I stood there, the familiar shape of a bridge appeared.

This example illustrates important features of the system. In other words, this system has a data structure. The graph representing the structure can be segmented by planes in any 4D direction.

This does not necessarily mean that it is aligned with any of the original coordinate planes.

With the result and j~, there are unexpected regularities in the data with the aid of visual perception. Sometimes things become clear. This can be seen in the “factor analysis method” of the dataset. This method can also be called a visual application version of the standard coordinate rotation method.

The same system is useful in fields such as exploring relationships in empirical data. It is applicable. Figure 26 shows the connections connected to each other at the input section 81 in the laboratory. shows a hypergraph device receiving input data from two resonant systems 80; ing. Although this equipment is shown here as an electrical wiring diagram, it is It can be a physical, chemical, or biological system, or an industrial It may be a process or a medical image forming apparatus.

In general, this is useful when considering collected datasets, regardless of the type of data. be. ,: This is a standard laboratory instrument that collects and processes data, e.g. Functions as a module in a quasi-laboratory storage oscilloscope. like this The data examined in the collected through the device, stored in any convenient database storage device 82, and 83 filtering, Fourier transform, and other existing techniques and logic. After processing, the data is output to an output unit 84 (computer monitor). So the data is Considered in arbitrary combinations and rotated in four dimensions, the interrelationships in the data set are , are considered from the position where they are best shown. sample of a set of 4 or more variables Pulling is performed using any combination of four parameters, and the desired model is created in module 83. It is possible to "parameterize" it with a sequence. Furthermore, the coordinates ξ, η are defined as There is also an option to output the image points of the captured data to the desired external output system. It is possible.

In this system, it is absolutely not necessary to treat input data as numerical values. Images in the form of data obtained by scanning or other possible methods are not supported by this system. In this way, it can be presented as a single image containing four dimensions, especially in alternating geometry. subordinate Therefore, the user has two cameras or other visual devices (for example, fM) on the four coordinate axes. You can view a collection of two-dimensional images obtained by a medical image forming device, etc.

Also, data on 4 channels as in the case of EKG or EEO recording devices. You can also consider the relationship. With practice, you can combine information at new levels. It is possible to visualize more complex information. “What’s around the corner,” so to speak? You can also see J. If we take time as the fourth coordinate axis, we can create a four-dimensional picture of the process. Imaging allows the motion of 3D objects to be seen in a single image.

C1 Geometric drawing/design “blackboard” mode from a mathematical perspective In other words, ``blackboard'' mode is unique in that the process begins with output. of The input definition of the object is drawn without leaving the "Chalk Tray". This is done by a type of command. The subsequent decision is to reverse ("upstream") the light in S0 Proceeding from a line to a ray in alternating geometry. The method shown in Figure 10 is originally Such a determination is also possible because the direction can be reversed. Just enough objects It is a condition that it be described in . Lack of sufficient description (common) The system makes arbitrary choices.

If the user issues any more commands to draw a plane in Euclidean 4D space, If we issue the specification for When you select a plane (plane) and input it to create that image, the system's eye position also changes. If there is no other information, the system defaults to alternating geometry. (3rd stage), and draw a standard and advantageous configuration in which all four coordinates can be clearly seen. The user invokes the cursor to draw a straight line between two points, but the Assume that it is not defined separately.

In this case, the system will place these two points on the plane If we request to draw a perpendicular line to a certain point on the plane obtained by Since there is not only one vertical line, the system places this vertical line in 30 and sets V to v. -0, the user would then set a given = one point (perhaps above) If we request a perpendicular line to So at point ), the problem becomes clear. , the system reacts by placing a line parallel to the V axis.

In all cases, whether specified by the user or not, the system will Assign four coordinates to the vector. The object that cannot be drawn is three-dimensional. Even if the for example, a request for a four-dimensional increment on the object). Again, the system The faster the system closes the loop between stage 8 and stage 4, the more responsive the drawing will be on the alternative medium. It is thought to be direct and give the feeling of actually painting. mouse and mosquito - Sol, light pen, touch screen, graphics pad, etc. The "mark" is the cursor position (actually the often used (ξ, η) output coordinates) and From there, position determination in the reverse direction is easy for those skilled in the field of 4. This is done in a manner that is self-explanatory upon reference to this description.

The output on the blackboard may be very elaborate, but for the purpose of examination it is It is sufficient to draw a frame. In this case, only important points are calculated and simple lines Connected by shape interpolation. When the operator performs the drawing using the normal draw tool J. You can also do that. If your computer system is not powerful enough, A simple procedure may help get a faster response. On the other hand, a large compilation For computer systems, highly developed, multifunctional and flexible CAD programs are required. You can work at full speed using the ram.

"Blackboard" acts as a virtual blackboard and displays various shapes and objects in alternating geometry. Figures 27(a)-27(e) are useful for constructing and studying the One of the control panels configured to implement the mode is shown. . The system shown here is a Matsukin Trash computer used to run this example. It was created using a hyper card to run on.

In FIG. 27(a), the "blackboard" is the window 55 on the computer screen 9. It is shown as This blackboard is a projection device that presents a visible image through electronic output. or any other medium. It is printed on the actual blackboard in the classroom. It utilizes available commercial technology that allows the projection of computer displays. This can be considered as a virtual blackboard as an output element of the system constituting the present invention. is the most appropriate. A virtual blackboard is not like a rigid three-dimensional Euclidean material. It moves in accordance with the postulates of alternating geometry, as if it were made of plastic. cormorant The window 55 is used to carry out the operation of the system in this mode. It is also used for certain display purposes. These even if a different display is given on the blackboard. will be displayed on the computer screen.

In this mode, the system provides drawing conveniences not available otherwise.

In most cases, the goal is to find a convenient and visible position to view the object. Therefore, it is sufficient to be able to manage the position and orientation of objects on the blackboard. child Therefore, the normal display on the blackboard is in analog format. Figure 27(a) and Figure 27( b) The q analog reading module 6o shown in the figure reads the X and y positions on the screen. 2. Read the angular position of the plane using the dial 62.

In the same way, the zv analog reading model shown in FIGS. 27(a) and 27(c) is Joule 65 connects the zv coordinate on the screen and the a plane on the dial 67. and read the corner position. xy analog reading module 60 toggle analog reading To switch between the pickup modules 65, press the buttons 63 and 68, respectively. This is done by

Object handler module 7 shown in Figure 27(a1) and Figure 27(d) 0 includes four buttons for left/right and up/down movements in this control. reading mode All modules are always under active control of this handler. Active mode Joule's lamp is on. Button 71 and button 72 are for continuous rotation during rotation. giving you control. Buttons 71 and 72 are used to control the continuation of rotation as before. When given a step and double-clicked, the managed object enters a continuous rotational motion. . Button 73 displays a list of objects under management in one corner of the blackboard when activated. A numerical reading section appears to indicate the current position and direction.

The chalk box module 75 shown in FIGS. 27(a) and 27(e) The transducer of the present invention can be used in selected geometries. Contains tools such as rulers and compasses for drawing various shapes. I'll show you now The Clid 4-dimensional space example has objects as seen in alternating geometry. Includes toolbox. Each of the buttons 76 calls one tool. used for. Regular computer-supported drawing or design Like tools used in design programs, these effectively define the postulates of the geometry in question. As a basis, it is possible to construct geometric figures in the relevant space. The present invention is based on this It makes it possible to systematically represent geometric shapes such as This makes such systematic drawing possible.

In the case of Euclidean 4D space, objects such as hypercubes can be directly represented. It is possible to express it. In this case, use a drawing tool to define the location. Each position is accepted as input to the second stage of the overall flowchart, and then Interpreted by the transducer in the fifth stage, then the corresponding image points are determined on the blackboard rays are calculated and displayed. Calling tools, marking points, directing between two points Connect by line, determine plane by third point, pass through plane, etc. The scheduler calls are made continuously and output is also produced. the system, Determine the intersection of 2,000 planes (at one point), as in "graph" mode There are also abilities such as lighting, shading, using color, specifying surfaces, and more.

For all of these, in line with the spirit of this explanation, computer drawings Directly apply standard technology of design and design to please those who are adept at this path. You can also create a chalk box tool kit with rulers and compasses. Objects drawn with can be selected, cut, copied, moved, and otherwise processed.

Only objects selected in this way will be added to the object handler module. Under the control of 70. A ruler by itself can draw a line between any two points. Ru. (i.e. specify a point for the transducer and You can get a line as the output of ) The lines on the drawing board can be drawn anywhere in four-dimensional space. It can also be placed in the direction. Any line or plane in that space can be rotated into the plane of the drawing board. can be visualized and manipulated.

Button 77 above the chalk box calls up an icon that identifies the button above the box. I can do it. On the other hand, button 78 is associated with one of buttons 76 and its tool. Calls up a menu that makes options available, the icon is displayed directly above the box. The menu position can be moved as a user option. It can be moved around and usually appears next to the blackboard. Menu, icons, and more on the blackboard The various shapes are arranged and managed in a format similar to that seen on a desktop computer display. good.

Enter information from external electronically outputtable sources and display it on the blackboard. and some system capable of receiving electronic information from the shapes created on the blackboard, Outputting to a storage divide or medium is done through normal file handling routines. In this way, the blackboard will be able to utilize the technology of graphic materials. Or for educational design, illustrations for books and magazine articles, or for videos and films. Helps in initial determination of object position.

1. Blackboard mode in 3D Lobachievsky geometry in non-Euclidean geometry It is impossible to always accurately depict objects in Euclidean space. For example, it is impossible to draw a simple Euclidean square with Robochevsky geometry. . In Figure 23(a), the user places Lobachevsky on the Euclidean 3D blackboard. - When trying to create a square based on geometry, the user first selects line segment AB. , one would draw a vertical line to A and B of equal length to AB. But the user When trying to make a vertical line CΩ, it is the user aiming to complete the square. I miss the mark.

Next, if the user extends 431) and tries to intersect with CD at point E, the second As shown in Figure 8(b), a quadrilateral is created on Lambel 1, and the angle CEB is a square. The angle is smaller than the right angle required for the shape.

The user can also force a straight line to be drawn between two points C and D. However, Ro In Baczewski geometry, line segment CD is not straight but curved. Therefore, Point angles ACD and CDB cannot be right angles, as shown in FIG. 28(c).

Since generally instead of a square the user is creating a 5accheri quadrilateral. be.

Thus the user is defined according to the fraction set (i.e. Lobachevsky geometry). It is possible to draw the plane shape on the surface of a different geometry (I!I, Euclidean geometry). I realized that this is impossible.

The present invention provides a system eye with conversion function to overcome the limitations of the human eye. do. The human eye is a Euclidean surface, and Figures 23(a) and 23(b) ) As shown in the figure, Lobachevsky objects can always be visualized accurately. The blackboard using the present invention is the eye plane of the transducer system. Since it mediates two different geometries, the blackboard is effectively a mouthpiece. Evski becomes one plane.

Figure 2'Ja) shows a normal Euclidean blackboard without the aid of the present invention. The tool is a CAD drawing system tool that can be used with a button on the chalk box. A Lamberj quadrilateral, not shown in Figure 29(a), was created using the following tool. but While the blackboard is in Euclidean mode, following the postulates of Euclidean geometry There is. As a result, to the unaided Euclidean eye, the line segment JF is It does not appear as a straight line, users can use this invention to change the mode of the blackboard -Can be turned into a relationship. The user's eyes then look at the transducer. Through the eyes of the system, we can understand the Lambertian quadrilateral in the space of Lobachevsky geometry. It can be visualized as it appears on the surface. The eyes of the system Lambertian quadrilateral as it appears in Lobachevsky space on the user's retina The rays constituting the image points corresponding to the rays from the object points are transmitted. the The results are displayed on the Lobachevsky blackboard shown in Figure 28(b). Figure 1O According to the process, the present invention transfers the quadrilateral object point to label 1 with its corresponding image points in normal eye space. The Lambertian quadrilateral is explained earlier in the book Among the “visualization and operation modes” of the invention, “go/see/touch” or “move” Visualize and manipulate on the blackboard according to the flow in the "turn/turn/touch" mode. It is an object that can be In “move/turn/touch” mode, the system is as described above. using the same mathematical flow, but now using trigonometry and 3D Lobatievsky geometry. It uses the postulate of Similarly, in the "Go/Touch" mode, the first player is 40 euros. The same mathematical flow as explained in the case of clit geometry, but now with Lobachevski. – Uses trigonometry and postulates.

For example, a selection with "blackboard" and "go/see/touch/mode selection (first stage)" 3D Lobatievsky geometry (second stage), the object As shown in Figure 30, it is in the form of a Lambertian quadrilateral with sides x, y, z, and m. defined (second stage), the transducer (fifth stage) - Angle of incidence of a ray from an object in 3D unified space using the relationship Determine.

α-2 dance’ (e”) α'-(90-α) m' --11 (mn (1'/2) ε = 2 yen l' (ex).

c　’　　　-(90-ε).

x’--In (tan e’f2’)s-cosh” (cash q cash It will be carried out as explained in "Go/See/Touch J Mode" inside.

When the complete CAD embodiment of the present invention is applied to geometric shape creation and drawing , the user selects an alternating geometry object from the chalk box and selects an alternating geometry object from the chalk box. When you put it on the screen inside the blackboard, you can see that the process is going on. For the sake of explanation, here are two points specified directly in the Lobachelaski plane. Let's follow the procedure for a concrete example of drawing a straight line in between.

After selecting “Lobachenosky Relationship”, the user selects the option shown in Figure 31(a). Specify points P and Q between which a straight line should be drawn. The selection of these two points is simply a matter of light sensitivity. by pointing on the computer screen or by clicking the mouse or by clicking the on a digital drawing tablet or in any of many other ways The transducer of the present invention operates to create a single image. do. The user can immediately view the results on the system placed in the 3D Lobatsky space. The image seen by the human eye can be seen as a line image passing through the two selected points. this The results can be seen as Figure 31(b). Here's what users experienced is the drawing of a line in the 30 Bacchievsky space, or the Lodyevsky space. This is what I learned from a blackboard object that behaves in a similar manner.

To accomplish this, the system performs the steps illustrated in the flowchart of Figure 32. pass. The first step is to use standard computer utilities to select Determine the screen coordinates of a point. These points are covered by the “chalk box” option described earlier. The results are shown in Figures 33(a) and 33(b). , in the 2nd and 3rd stages, Lobachevsky coordinates (x, y) are first set on the retina or on the image plane. We have described the position determination by moving to the coordinates (ξ, η), but the process (Fig. 10) flow) is reversed and, using the appropriate trigonometry described above, the selected stream is The clean coordinates are the same as those already mentioned in section 3A, 2 [Go/See/Touch]. Move to Baczewski coordinates J (X, y). Again, Figure 33(c) and Figure 33 (d) Switch to "Euclidean blackboard" mode to display the results shown in the figure It is possible, and it explains our purpose, but in normal CAD use usually doesn't need to do that.

The system performs an algebraic test in the fourth stage, and the two points are Lambertian quadrilaterals. Determine whether it is on the line forming the root J. In some procedures this is shown in Figure 33. This is done by determining whether the sign of the discriminant in the curve fitting equation in the fifth stage is positive or negative. Positive If the sign of Lobachevsky coordinates of the roof of a suitable Lambertian quadrilateral using the method Determine the formula. Conversely, if the discriminant is negative, the required line must intersect the baseline. Dew. In this case, the program moves to Section 6(b) ``Q4Z,'' and sets these points as part of the baseline. Fit one side of the Lobachevsky right triangle with as one side. same Even in the case of Both must be calculated. If you want, you can use the blackboard in "Euclidean" mode. , and plot this line as it appears on the Euclidean plane as shown in Figure 33(d). You may.

In the 8th stage, this line is taken as an object in Lobachevsky space. , the transducer of the present invention enables the "go/see/touch" mode in Section A.1. moved to epiretinal coordinates in the manner already described for the Plotted on the board as shown in Figure 33(b). The points given on the blackboard An image of the passing line is created as shown in Figure 33(b). What is shown here is the original This embodiment is not intended to limit the invention in any way. It's not.

Generally speaking, CAD users understand the background process or method just described. Note that the intermediate "Euclidean background" image reproduced for illustrative purposes is I would like to emphasize that there is no way for a user to draw a line in Lobachevsky space. I only have direct experience of it being pulled. The user selects “Ruler J” from the chalk box. After making your selection, you can simply point to the selected point and watch the desired line appear. Wear. A straight line is used here as a basic example to explain the blackboard method for alternating geometry. However, any geometric shape or design can be It can also be implemented by applying the same technology. For example, a figure consisting of a curve is , using any number of linear elements, or other computer-generated curve drawings. The use of "splints" often used in drawing or computer graphics systems (s (Printing) allows the approximation to be as accurate as desired. this Existing CAD technology that enhances and assists drawing and design processes such as It can be extended within the scope of the technique and applied to solve this problem.

A pair of points chosen for convenience of explanation produced a "negative" result in the fifth row of FIG. 32. too If this value is positive, the system places these points on the [roof] of the Lambertian quadrilateral. and found that 1-set of points are located along the line. This is chosen to produce a straight line in the Lobachevsky space. When these points are plotted on the Euclidean plane, we can draw a curve on the Euclidean plane. It is entirely possible that this will occur. In Figures 30(a) and 30(b) above, , draw a straight line in the Lobachevski plane and do so in the Euclidean plane. We will explain the differences between the Czewski showed how much meaning it has when drawn on a single plane. already explained We extend the technique of drawing on a plane using the technique described above, and draw 3 on the Lobachevsky surface. It is possible to design dimensional features.

The present invention thus provides a design and imaging system for objects in alternating geometry. provide a program. For those who are technically proficient in this field. Considering the things described above, Various modifications and adaptations of the invention may be made in light of the above considerations. For example, one shot The light is limited to 4D Euclidean geometry and 3D Lobatievsky geometry. There isn't. This system uses Riemann's 3D curve system (differential geometry), high dimensionality, It can be applied to other cases as well. Object drawing in alternating geometry of the present invention The transducer of the present invention can be used in imaging systems in modes other than those described above. It can be used as long as it is commercially available. This system allows display, video, hardcopy output as a movie, spleoptic display, stereographic display, It is possible to create spreophonic output, etc.

Figure 2 □ 180' 150' 120 115' 97.5' 90'□ Figure 5a Figure 5 Figure 6 Figure 7 ′　2 Observer's eyes [Retinal surface Figure 11a Figure 15 Figure 18a Figure 18b Figure 13c Figure 19a Figure 19d Figure 20a Figure 20c Figure 20e Figure 20f Figure 20g Figure 20h +y Figure 22d 1 Figure 23b Figure 23a Figure 23d Figure 27a Figure 27e Figure 28a Figure 28b Figure 28e Figure 29a Figure 29b Figure 30 Figure 31a Figure 31b Figure 33a Figure 33e Procedural amendment (acid) 1.Display of the incident pCT/US91108919 1992 Patent Application No. 502943 2, Title of Invention Object image creation system in alternating geometry 3, corrector Relationship to the incident: Patent applicant Address: Prince George, Annapolis, Maryland 41404, United States George Street 213 Name Balaspectives Incoholated Representative Simpson, Thomas Kay (Nationality) (United States of America) 6. Contents of amendment international search report 1+Ir groan 116Am&mf #4 to 8-r ha! ;Ql/nl'IQIQ Freon Continuation of top page (81) Designated countries EP (AT, BE, CH, DE.

DK, ES, FR, GB, GR, IT, LU, NL, SE), 0A (Bze, BJ , CF, CG, CI, CM, GA, GN, ML, MR, SN, TD, TG), A T, AU, BB, BG, BR, CA, CH, C3, DE, DK.

ES, FI, GB, HU, JP, KP, KR, LK, LU, MC, MG, MN, MW, NL, No, PL, R○, SD, SE, SU

Claims (30)

[Claims] 1. An object image forming system in alternating geometry, the system comprising: and an input device for inputting objects and selected objects in alternating geometry. A 2D image in the user's 3D Euclidean space is created by converting the light rays from the object point. a transducer for making an image point; Assemble image points to create an object that is directly visible to the user's eye and placed in an alternating geometry. a processing device for forming an image of the object and for assembling and transforming the object; an output device for presenting the object to the user's eyes; and an alternating geometry on the user's human retina in three-dimensional Euclidean space. Objects in alternating geometry characterized by directly forming an object image of image forming system. 2. A patent claim characterized in that the alternating geometry is a four-dimensional Euclidean geometry. The system described in Scope 1. 3. A patent claim characterized in that the alternating geometry is 3D Lobachevsky geometry The system according to item 1. 4. characterized in that the input device includes inputting a coordinate system for alternating geometry; A system according to claim 1. 5. Claims characterized in that the input device includes selecting an alternating geometry. The system described in item 1 below. 6. The object device is a hand for interactive manipulation of objects in alternating geometry. The system of claim 1 further comprising a stage. 7. Object operations allow objects to perform plane rotation and translation. The system according to claim 6, characterized in that it means that the system can be stem. 8. The input device further provides a means for inputting the user's observation position and the direction of the user's line of sight axis. A system according to claim 1, characterized in that the system comprises: 9. The input device further includes means for interactively manipulating the user's observation position and line of sight axis. 9. The system of claim 8. 10. characterized in that the interactive operating means can perform plane rotation and translation of the user; A system according to claim 9. 11. the input device includes an active button on the screen of a computer monitor The system according to claim 1, characterized in that: 12. Patent characterized in that the output device is a computer monitor screen The system according to claim 1. 13. A patent claim characterized in that an object in alternating geometry is a graph The system described in Scope 1. 14. The system is characterized by graphing one object in alternating geometry The system according to claim 1. 15. Claim 15-, characterized in that the object is a complex function. System described in section. 16. Claim 16, characterized in that the complex function is a hyperbolic cosine. System described in section. 17. The input device represents the data structure using a plane in an arbitrary four-dimensional orientation. Claim 15, further comprising means for segmenting the graph. system. 18. The system is characterized by composing and drawing objects in alternating geometry. A system according to claim 1. 19. A method of creating an image of an object in alternating geometry, the method comprising: Define objects with geometric properties, user position and orientation of the user's line of sight rays from the object point to the 3D Euclidean sky where the user is located. Convert to corresponding two-dimensional image points in between, Assemble image points to construct the direct appearance of objects in alternating geometry, The user is placed and oriented in an alternating geometry with objects relative to the user. An object in alternating geometry consists of presenting it as if it were directly visible. Creates an image of the object on the user's human retina in 3D Euclidean space A method for creating an image of one object in alternating geometry, characterized by: 20. The method moves objects interactively in alternating geometry and further comprising a procedural step of interactively rotating the object about a plane. 20. The method according to claim 19, characterized in that: 21. The method is based on the movement of the user's position and the orientation of the user's gaze axis during alternating geometry. Claims further comprising a procedural step of interactive attribution of plane rotations The method according to paragraph 19. 22. The step of defining the properties and objects of alternating geometry is the coordinate system in alternating geometry. 20. A method according to claim 19, characterized in that it includes the definition of . 23. The process of defining alternating geometry properties and objects interactively Claim 19 includes specifying the position and orientation of Method described. 24. The stages of defining alternating geometry properties and objects are interactively guided by the user's position. and specifying the location and orientation of the user's line of sight axis. The method according to scope item 19. 25. The transformation stage is where the rays from the object points of the object are D The angle of incidence when entering Euclidean space is determined using the geometric relationships of alternating geometry. A patent claim characterized in that it includes the step of determining the x-axis and the y-axis using the The method according to item 19. 26. The transformation step is from the angle of incidence of the ray at the object point to the three-dimensional Euclidean Equivalent angles with respect to the x and y axes in space can be expressed as three-dimensional Euclidean geometry. Claim 25, characterized in that it includes a step of determining using various relationships. the method of. 27. In the method of claim 25, the step of determining From the angle of the object point in the grid space, we can calculate the geometry of three-dimensional Euclidean geometry. The patent further comprises the step of determining two-dimensional image points using various relationships. The method of claim 25. 28. The method is a method of graphing the appearance of objects in alternating geometry. 20. A method according to claim 19, characterized in that: 29. The method is a method of geometric construction of an object in alternating geometry. 20. The method of claim 19, characterized in that: 30. characterized in that the procedural steps of the method are performed in the order stated in the specification. 19. The method of claim 19.