JPH02109183A - Free curve generating method - Google Patents

Abstract

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

Description

[Detailed description of the invention] A. Industrial field of application B. Outline of the invention C. Prior art Problems to be solved by the invention E. Means for solving the problems (Figures 1 and 4) F. Effects (Figures 1 and 4) G Example (G1) Principle of free curve (Figure 2) (G2) Example of free curve creation (Figures 1 to 4) (G
3) Other embodiments H Effects of the invention A Field of industrial application The present invention relates to a free curve creation method, for example CAD (c
This method is suitable for use when deforming the shape of a free curve generated in computer aided design.

3. Detailed Description of the Invention B Summary of the Invention The present invention provides a free curve creation method in which a predetermined 1ffi
Control to pass the overpoint? By generating the ajm vector, the original free curve can be deformed with a simple operation.

C Conventional technology For example, when designing the shape of an object with a free-form surface using a CAD method (geometric mode
In general, a designer specifies multiple points (these are called nodes) in a three-dimensional space through which a curved surface should pass, and uses a predetermined vector function to create a boundary curve network that connects the specified nodes. A curved surface expressed in a so-called wire frame is created by calculating the . In this way, it is possible to form a large number of framework spaces surrounded by boundary curves (such processing is hereinafter referred to as framework processing).

The boundary curve network formed by such framework processing itself represents the rough shape that the designer intends to design, and a curved surface that can be expressed by a predetermined vector function is calculated using interpolation using the boundary curves surrounding each framework space. If it is possible to do so, it is possible to generate a free-form surface (which cannot be defined by a quadratic function) that is designed by the designer as a whole. Here, the curved surfaces stretched in each framework space form basic elements that constitute the entire curved surface, and these are called batches.

Conventionally, in this type of CAD system, as a vector function expressing a boundary curve network, easy-to-calculate vector functions such as Bezrer equation and B-spline (B-Spline) are used.
A third-order tensor product consisting of the formula 1ne) is used,
For example, it is considered to be optimal for mathematically expressing free-form surfaces that have no special features in shape.

In other words, a free-form surface that has no special features in shape is
When points given in space are projected onto the xy plane, the projected points are often arranged regularly in a matrix, and when the number of projected points is expressed as mXn, the framework space It is known that it can be easily stretched using a quadrilateral patch expressed by a cubic Bezier equation.

However, when this conventional mathematical expression is applied to a curved surface with a characteristic shape (for example, a curved surface with a greatly distorted shape), it is difficult to connect batches together, and it requires advanced mathematical calculation processing. Since it is necessary to execute the method, there is a problem that the calculation processing by the computer becomes complicated and enormous, and the calculation time becomes long.

As a way to solve this problem, we find internal control points that satisfy the condition of tangent plane continuity for the shared boundaries of adjacent framework spaces, and use a vector function that represents the free-form surface determined by the internal control points to A method of attaching a patch with a curved surface has been proposed (Patent Application No. 60-277).
No. 448, Patent Application No. 1984-290849, Patent Application No. 1983-
No. 298638, Japanese Patent Application No. 15396, 1983, Japanese Patent Application No. 1983
1-33412, Japanese Patent Application No. 61-59790, Japanese Patent Application No. 61-64560, Japanese Patent Application No. 61-96368, Japanese Patent Application No. 61-69385).

Problems that the invention aims to solve By the way, in reality, designers' design work usually involves repeating local corrections to move step by step closer to the shape that the designer envisions. The free-form surface generated as described above is also repeatedly modified to approximate the shape envisioned by the designer.

In other words, as a method for modifying such a free-form surface, the designer cuts the free-form surface at a desired cross section on the display screen, visually confirms the cross-sectional shape of the resulting free-form surface, and locally corrects the details. After that, a method is used to generate a free-form surface again based on the cross-sectional shape (
Patent Application No. 1982-276925, Patent Application No. 1982-27869
No. 4).

However, when a designer uses such a method to modify a free-form surface, he or she may wish to modify the original cross-sectional shape so that it passes through a predetermined point (hereinafter referred to as a passing point).

In such cases, conventionally, the cross-sectional shape of the curve is represented by -6~ For each curve segment of the free curve, the nodes and internal control points are moved in arbitrary directions to transform the curve shape for each curve segment. , the free curve passing through the desired passing point must be regenerated by reconnecting the adjacent curve segment, which poses a problem that requires extremely complicated design work for the designer.

The present invention has been made in consideration of the above points, and aims to propose a method for creating a free curve that can transform an original free curve so that it passes through a desired passing point i1 through a simple operation. It is.

E Means for Solving the Problem In order to solve this problem, in the present invention, the first and second nodes P0 and P3, and the first and second node P0
Using the first and second control points P and P2 set between P5 and P5, a free curve expressed by a predetermined vector function R([) is added from the first node P0 to the first A tangent vector a toward the control point P1 is extended to generate a control line vector PIN-p, and the control line vector PIN
New control points p, N determined by PO, and the first and second
By generating a new free curve expressed using the nodes Po and P3 and the second control point P2, a new free curve K3GN passing through the desired passing point P7 is obtained.

F action first node P. If the tangent vector a directed from to the first control point P is extended to generate the control line vector P+wPo, one of the two control line vectors, the control line vector P can be easily I N'-P O
A control point PIN determined by is obtained, which allows the free curve to pass through a predetermined passing point P. can be expressed.

G example An embodiment of the present invention will be described in detail below with reference to the drawings.

(G1) Principle of free curve In this example, the free curve is as shown in Figure 2.
Nodes P set at arbitrary intervals. and Pz(P+.,
. and P, 3. . ,P. , I and P 1311, P.
. it and Pn+z) are connected.

This curve segment KsG is a cubic Bezier (bezi)
er) using the following formula, R(t)=(1,-t+tE)"Pa -old-(1
) is expressed by a parametric spatial curve R(t).

Here, t is the curve segment Ksr from one node P0
It is a parameter that changes from a value of 0 to a value of 1 as expressed by the following equation (2) in the direction along the direction until reaching the other node P.

In this way, in the curve segment represented by the cubic Bezier equation, G is set to the nodes P0 and 23 by the shift operator.
By specifying two control points P1 and P2 between them, each point on the curve segment KSG is expressed as a position vector R(t) from the origin O in the xyz space by the following expansion formula.

Here, the shift operator E has a relationship between the curve segment and the control point P on G as expressed by the following equation. Therefore, by expanding equation (1) and substituting the relationship in equation (4), we get the following equation R(t)-(1 t+tE)''P.

1 〇- -((],, -t)3+3 (1-1)" t E+
3(1-t)t”E” 10t”E 3) Pa −(1-t)3po+3(1~t)”tEP0+3(1
-1)t”E2Po +t3E3p0=(1-1)'P
o+3(1-1)"LP++3(1-t)t2P2+
t'p.

・・・・・・ It can be calculated as shown in (6), and the result is (Formula 32%Formula%) Thus, each curve segment Ksc, o, K on the free curve
sa+ % Ksaz is expressed as (3) &, [Zite Sore Sore two nodes and control point P,. ,. ~2, P C
11°~2, Pai+. ~2 and P. 10 to 2, each curve segment Ksco −, Ks
Each curve segment KSG by representing the free curve using c+, Kscz, respectively. , KsG+ ,
Nodes and control points Pco+oz, P expressing KsGt
(110-2, P(210yZ and P(310-2) can be used to express a free curve.

(G2) Example of free curve creation In this example, the central processing unit (CPtJ') of the free curve creation device executes the curve deformation processing program shown in Fig. 1 to pass through the specified input passing point. like,
A desired curve segment is deformed, thereby obtaining a free curve made up of the curve segment and deformed so as to pass through a passing point.

That is, the CPU of the free curve creation device performs step SP
The process moves from I to step SP2 and waits for the designation of a curve segment to be transformed.

Here, as shown in Fig. 2, in the free curve, common nodes P (0) l (P tau o), P+
o+z(PI3)l>, for example, three curve segments connected such that their tangents coincide) KSGO, K
SCI, K! 162, so that the curve shape of the entire free curve changes smoothly.

In the free curve creation device, when the designer uses a mouse, for example, to move the cursor over the first curve segment KsGt and then clicks the mouse, the input of the curve segment to be deformed is received, and then step SP
3 and waits for input of passing point P1.

At this time, as shown in FIG. 3, in the CPU, 3G+ nodes P0 and P and control points p,
After coordinate transformation so that the plane passing through , p becomes the xy plane, the curve segment KsGl is transformed into nodes P, , P3
and control points PI and Pt.

Furthermore, the node P. Obtain the intersection point QX of the straight line extending from to the control point P, and the straight line extending from the node P to the control point P2, and draw the line segments L1 and L2 connecting the intersection Qx and the nodes Po, P:+ as a curved line. Display 3G+ on the segment at the same time.

In practice, a curve segment Ks expressed by a cubic Bezier equation
When deforming G+ by resetting control points P+ and Pz, the straight line extending from node P0 to control point P, and the straight line extending from node P8 to control point P2, become nodes P0 and I', l, respectively. , the adjacent curve segment K11(i.

In order to transform , , into a curved segment while maintaining the state in which the tangent line coincides with Ksa□ and Ksa□, it is necessary to reset the control point on the straight line.

Furthermore, at this time, on the straight line extending from the node P0 to the control point P1 and the straight line extending from the node P to the control point P2, if the MIl1 point is reset to a line segment and a straight line other than Lx, respectively, There is a problem that the curve segment expressed by the reset control points may draw a loop, so in order to perform deformation processing while maintaining the rough shape of the curve segment (sGt), it is necessary to reset the control points. The position is limited to and above the line segment L+.

Therefore, in the curve segment) KSGI, the node P0
The deformation process may be performed so as to protrude from the triangular area connecting P3 and the intersection point Qx.In the end, the position where the passing point Pt is designated and input is limited to the triangular area.

Thus, in the designer, the curve segment hKs
The passing point p at which G+ is transformed is the corresponding line segment Ll and L2
This can simplify the transformation process.

CP 1. When the passing point Pa is input, the controller J then proceeds to step SP4 and resets the control points so that the curve segment KsG+ passes through the passing point pt.

As shown in FIG. 4, in this embodiment, the vector PIPa (that is, the tangent vector at the node P., hereinafter referred to as the control line vector of the node P.) directed from the node P0 to the control point P is extended; A control point PIN is set at a point determined by the control line vector PINFO.

As a result, the control points PI and I are reset on the line segment L1 for one control point P among the two control points PI and P2, and the adjacent curve segment Ksc is set again.

and a node P shared with 6□. and P3, the curve segment Ksc+ is deformed so as to pass through the passing point PT while maintaining the state in which the tangents coincide.

In this case, the newly set control point P1N is calculated by the following formula %, where the control line vector P IP o of the node P0 is set as the jB-order vector as a, and the extension amount for the unit vector a is expressed using the scalar quantity uc. Equation % (7) Therefore, the curve segment (K3GN) represented by the control points P, N, the nodes P, and P2, and the control point P2 is expressed as follows: instead of the parameter t, the parameter uc that changes between the values θ and From equation (3), using equation (7), the following equation R(u,+ue)=(tu,)''P.

+3 (1ui)”us (PO+a,・uC)+3
(1us) uiz Pz +u, 3 P3 ...... (8) It is expressed as follows.

Therefore, in order for the curve segment K3ON to pass through the passing point P1, the following equation % (9) only needs to hold, and if a scalar quantity uc that satisfies this relationship is obtained, then from equation (7) Coordinate data of a new control point PIN can be obtained.

By the way, in practice, when obtaining coordinate data in a three-dimensional space from equation (9), a nonlinear cubic equation must be solved,
When performing this type of arithmetic processing, there is a problem in that the processing time becomes long due to repeated calculations.

Therefore, in this embodiment, in step SP2, the coordinates of the y-axis and the Obtain only coordinate data.

Furthermore, by linearizing equation (9) and performing arithmetic processing using Newton-Labson's method, coordinate data can be obtained within a short time.

In other words, by performing Taylor expansion on equation (9) and eliminating its quadratic and subsequent terms, the following equation (%) is obtained, and by transforming this, the following equation -P t R (u * + u c) is obtained.

Therefore, if formula (II) is decomposed into y-axis and y-axis components and each component is shown with a subscript, the following formula +3(1, -u,)11. ” Pz +u, 'P3.

Here, for each component of the matrix, from (8), the following formula % formula %) Thus, formulas (13), (14), and (]5) are decomposed into X and y axis components to form formula (12). After the assignment, initial values are assigned to U and uc, and sequentially Δu1 and Δuc
Substitute.

This allows ΔU and Δuc to converge equation (12)
The scalar quantity uc can be obtained from the value ΔU, Δuc, U, and the initial value of uc.

Thus, based on the scalar quantity uc, a new control point P
By setting IN and obtaining 3GN for the curve segment represented by the control point PIN, control point Pt, and nodes Po and Ps, it is possible to obtain a free curve that passes through the designated input passing point Pa.

By the way, for both the two control points P1 and P2,
A method of deforming the curve segment by resetting the control points and using the passing points PT is also considered, but in practice, this method has the problem that it is difficult to unambiguously determine the control points to be reset. By resetting only one control point Pl out of the two control points P1 and Pt as in this embodiment, a free curve passing through the passing point P can be easily and reliably obtained.

When the new control point P1N is obtained, the CPU sets the new control point PIN% control point P2 and node P. , P3 into coordinate data on the original three-dimensional space.

At the same time, the deformed curve segment) KSGN represented by the control point PIN is converted to new control points P, N, control point P
2 and node P. , Ps, and then step SP
The process moves to step 5 to end the transformation processing procedure.

According to the above method, by expressing a curve segment with a control point determined by an extended control line vector, one of the curve segments connected so that the tangent lines coincide with each other at a common node can be The free curve can be transformed so as to pass through the designated passing point while maintaining the conditions for the curve, and thus the free curve can be transformed so as to pass through the desired passing point with a simple operation.

(G3) Other embodiments (1) In the above embodiments, a case has been described in which a curve segment expressed by a cubic Bezier equation stretched on an arbitrary plane is deformed. is not limited to this, and may be 4th or higher.

=21 (2) Furthermore, in the above embodiment, one of the curved segments connected such that the tangents coincide at the shared node
Although the present invention is not limited to this case, the present invention is not limited to this, and when deforming one of the connected curve segments with mismatched tangents,
Furthermore, it can be widely applied when deforming independent curve segments.

(3) Furthermore, in the above embodiment, a case was described in which a free curve is deformed so that it passes through a specified input passing point, but the present invention is not limited to this, and the present invention is not limited to this, and the present invention is not limited to this, but the present invention It can be widely applied to cases such as deformation processing.

H Effects of the Invention As described above, according to the present invention, by expressing a curve segment with control points determined by an extended control line vector, the curve segment can be transformed so as to pass through a predetermined passing point with a simple operation. Thus, a free curve passing through the passing point can be obtained.

[Brief explanation of the drawing]

FIG. 1 is a flowchart showing one embodiment of the present invention, and FIG.
The figure is a route map for explaining the curved segment, and FIGS. 3 and 4 are route maps for explaining the deformation of the first curved segment. Ksa+, KsGz, K11G3, KSGN...
...Curve segment, P(.H ~ 8, PL 3rd ~ 3
, P(.1N -48%P (3128+l)l'・
...Node, P(Ill-3, P C!l 1-3,
P(111)l... Control point.

Claims (1)

[Claims] In a free curve expressed by a predetermined vector function using first and second nodes and first and second control points set between the first and second nodes. , Generate a control line vector by extending a tangent vector from the first node to the first control point, and generate a new control point determined by the control line vector, the first and second nodes, and the control line vector. A method for creating a free curve, characterized in that the new free curve that passes through a desired passing point is obtained by generating a new free curve expressed using second control points.