JPH01120603A - Numerically controlled working method - Google Patents

Abstract

PURPOSE:To produce the working information effectively and at a high speed by forming a 3rd Bezier curve passing the given point groups into a straight line with no intervention of a B-spline curve. CONSTITUTION:The pass points P1-P4 are obtained from the data on the three- dimensional position vectors and chord vectors P2P1 going to a point P1 from a point P2 is normalized into a vector (a). In the same way, a chord vector P2P3 going to a point P3 from the point P2 is normalized into a vector (b). Furthermore the sum of both vectors (a) and (b) is defined as a vector n1. Then the vector n1 is used as a normal line vector and a plane passing the point P2 is referred to as pi1. In the same way, a normal vector n2 is obtained with pass points P2-P4 and a plane passing the point P3 is referred to as pi2. Then a plane passing points P1-P3 is referred to as pi3 and an intersecting point Q2 is obtained. A secondary Bezier curve C2 is produced with such pass points as P2, Q2, P3, etc., defined as control points. Then the degree of the curve C3 is increased up to a three-degree from a two-degree and the control points Q2' and Q2'' are obtained for a 3rd Bezier curve.

Description

[Detailed Description of the Invention] [Industrial Application Field] The present invention relates to a numerically controlled machining method, and is particularly suitable for machining the contour of a workpiece along a three-dimensional curve passing through a given point group. This paper relates to a numerically controlled machining method.

[Summary of the invention]

From the information of three successive points of the four given passing points, determine a plane that passes through the central passing point, find a point on that plane that is determined by the information of the other three passing points, and connect this point and the two passing points. A quadratic Bezier curve is generated using the and as control points, and by increasing the degree to the third degree, a cubic Bezier curve passing through four points is generated, which enables high-speed generation of machining information. , is a numerical processing method that facilitates local changes in passing points and improves design efficiency.

[Conventional technology]

A CAD/CAM system that handles three-dimensional free-form surface data inside a computer and generates NC data (tool path data) for automatically machining final products or molds using NC machine tools, etc. is now in practical use. It is becoming more and more popular.

When handling curved surfaces such as product external shapes in a computer, it is desirable to have properties that are favorable for design, such as good shape controllability (easy deformation and modification) and easy calculation. A three-dimensional model is represented as a set of surface elements (punch) made up of line elements that can be calculated using these formulas.

A line element is generally defined as a three-dimensional free curve consisting of a Bezier curve generated by specifying passing points.

Conventionally, such a Bezier curve has been created by first generating a B-spline curve that passes through a given point group, and then determining the control points of the Bezier curve from the control points of the B-spline curve.

[Problem that the invention seeks to solve]

In a CAD system, local control of a model is
This is extremely important in creating an interactive interface.

For example, when designing the cross-sectional shape of a workpiece as shown in Fig. 7, a group of points P, SP, -・-・−・・・ is given, and a Bezier curve passing through all of these points is created. generate.

However, in the secondary calculation results, the machining contour line may become depressed as shown in the X part of the figure. Further, the vertical and horizontal dimensions, A and B, may deviate from the given design values. In such a case, some of the points are reset and the outline is recalculated to obtain the desired outline.

However, the method of generating a Bezier curve using the B-spline curve as described above requires an extremely large amount of calculation to locally change the curve, and has the disadvantage that it is substantially difficult to make partial corrections.

As an example in FIG. 8, when passing points P0, Pl-・----111--・Ph (each three-dimensional position vector) are given, the B-spline curve passing through each point is The control point Q is defined by the -order simultaneous equations by the control points Q, -.

A control point vector Q+-Qs (indicated by white circles) is obtained from these equations, and then the control points are connected with straight lines and divided into three to calculate control points (indicated by black circles) of the Bezier curve.

Therefore, if we locally change the passing point like p, /, the mouth, ~
Ω, must be recalculated.

Furthermore, in order to generate a Bezier curve, control points of the B-spline curve must be found in the middle, which is inefficient. Although the Naori-spline curve has the feature of continuous curvature, in the case of three-dimensional modeling, the condition of tangential plane continuity is given, so generating a B-spline curve has little practical value.

In view of this problem, it is an object of the present invention to provide a numerical processing data generation method that can directly generate a Bezier curve passing through a given point group, and therefore makes it extremely easy to locally change the outline of a workpiece. shall be.

[Means for solving problems]

In order to determine the machining curve for numerically controlled machining, a group of passing points is first given.

Next, take out 4 consecutive points Pl-Pa and 3 of them
Regarding points P2 to P#, a plane π that has a normal vector determined based on information on two chord vectors from the center point P toward points on both sides and passes through point P1.

Establish.

Next, P! A straight line 12 passing through is determined.

Next, find the intersection point Qt of the above straight 'sit and the above plane π2, generate a quadratic Bezier curve with points P8, Q+, and Ps as control points, and transform this to a cubic degree to form a cubic Bezier curve. generate.

Numerical control machining data is obtained using the curve thus generated as the contour line of the workpiece.

[Effect]

The plane π2 is determined by the information of the three points pt-p, and the straight line lt is determined by the information of the three points PI''''P3. Therefore, a Bezier curve passing through each point is directly generated based on the information of the four consecutive points P1 to P4.

〔Example〕

FIG. 1 shows a method for generating a first embodiment of a Bezier curve passing through a given point group, and FIG. 2 shows a flowchart.

This first method is characterized in that the intersection of three planes is used as a control point of a quadratic Bezier curve, and the order is increased from quadratic to cubic through algebraic calculations.

In Figure 1, the given passing points are p, , p, , P
l, P4, and is given as three-dimensional position vector data. First, as shown in FIG. 2, in step S1, the string vector PtP+ directed from point P2 to Pl is normalized to vector a. Similarly, the string vector PtPs going from point P2 to P is normalized and set as vector b. Furthermore, let the sum of vectors a and b be vector n. For this vector n, the zero-order vector n1 located in the middle of vectors a and b is taken as a normal vector, and the plane passing through p is set as π1 (step S2).

Similarly, step s1 for passing points Pt-Ps, P4
, S2, the normal vector n2 is obtained, and the plane passing through the point P3 is defined as π2 (steps S3, S4). Next, in step S5, the plane passing through the points P1, P2, and P is set to π, and the intersection of the planes π3, π8, and π is determined as mouth 2 (
step S6), then in step S7, P2, Q2, h
A quadratic Bezier curve C2 with (P2, P, is an end point) as a control point is generated, and in step S8, the order of the curve c2 is increased from quadratic to cubic, and the control point 02' of the cubic Bezier curve is Q
Get t#.

When the number of given passing points is n, the subscript 1.2.3 in the explanation of Fig. 2 is changed to -1, va, nol and 1.
．． If the process is repeated for -8 to 7.8, a continuous Bezier curve Ct''C, -z passing through each passing point can be obtained.

Regarding the processing of the end point P, please refer to step S9 in FIG.
First, give an arbitrary tangential vector vI at P, 0
Next, in step SIO, the intersection Q1 between the extension line of the vector vl and the plane π obtained in step S2 is determined. Next, in step Sll, a quadratic Bezier curve c1 is generated with Pl, Pt (end points) and Q+ as control points, and in the next step S12, the quadratic to cubic curve is converted to define a cubic Bezier curve. new control points Q+', Q+.'' are calculated.

The same is true for the other end point P, (,=4), and step S
At step 13, the tangent vector Vt is given to P4, and the same processing as steps 39 to S12 is performed. That is, the plane π11-1
Let the intersection Q1%-1 of the extension line 12 of and V be the control point,
A Bezier curve CM-1 at the terminal end is obtained and the order is converted.

The shape of the curve does not change even if a quadratic Bezier curve is transformed into a cubic one by arithmetic operations.The proof of this is as follows.

As shown in Fig. 3, Po given in three-dimensional space,
A Bezier curve represented by three control point vectors consisting of Pt (endpoint) and P, is R(t) = (1-
t + tH) ”Po −−−−−−−−−(1
). t is a parameter that takes a value of O to 1 between both end points. Further, E is a shift operator indicating each control point, and is P+"EPo, Pg-E"P+.

Similarly, the cubic Bezier curve is R(t)-(1-t+tE)"P.

=(1-'t)3po+3(1-t)"EPG+3
(1-t) t "E"PO+t"E'po...
−・Represented by (2). Po, EPO, E Naka. ,E”P
In Fig. 3, o is the four control points P0 of the cubic Bezier curve,
Corresponding to Ql, Q2, and P2, respectively (f!Po −Q+,
tH"Po=Qz, E'Po-Pg).

Multiplying both sides of the first equation by (1-t) +t, -1, we get
((1-t) + t) R(t) = ((1t)+t)f(1t)”P0+ 2(1t)
tP+ +t”Pz)=(1-t)3P6+(1-t)
t(2Pl +P(1) t3P! ・−・−・
...-River---(31. Therefore, the second and third equations
If the expressions are equal, then ・−・−・−−−−−・−・−・(4), that is,
As shown in FIG. 3, if the line segment pop+ is divided proportionally at a ratio of 2:1, the control point Q is found, and the line segments p, p.

The control point Q2 can be found by proportionally dividing 2:1. The cubic Bezier curve determined by the four control points P0 and QiQiPt found in this way is the three control points P0 and P2.
, Pl is the same as a quadratic Bezier curve.

Next, Fig. 4 shows a second method of generating a Bezier curve passing through a given point group, and Fig. 5 shows a flowchart of the procedure. The method is characterized in that control points of a quadratic Bezier curve are generated and converted into a cubic curve.

First, as shown in FIG. 5, in step S1, point Pt5P
1. Regarding P4, let a2 be a vector that passes through the intermediate point P3 and is parallel to the string vector hP+ of the point P4 on both sides.

Next, in step S2, the cross product of the chord vectors hPz and P3P# of P:, Pi as seen from P is taken as a vector b.

This vector is a plane π2 passing through three points Pl, P3, and P9
is the normal vector of

Next, in step S3, take the cross product of vectors a and b2,
P3 in step S4 is normalized and becomes vector h.
Let π3 be a plane that passes through , and has n as its normal vector.

This plane π8 passes through P, is parallel to the chord P2P4, and is orthogonal to the plane passing through P2, Pl, and P#.
Next, in step S5, P! , the chord vector P.

A vector a1 parallel to the 1 year old is found, and the intersection Qz of the extension line 18 of a and the degree plane π2 is found. Hereinafter, as in the first method, a quadratic Bezier curve C with Pl, P3, and Q8 as control points
t, increase the order from 2nd to 3rd and control point 08
', Q□' are determined (steps S6, S7).

When n passing points are given, Pl-1, P, ,
Regarding P, +, the processing of 31 to S7 is performed as =2 to . ．． -2, a continuous Bezier curve is obtained.

Regarding the Bezier piece wACI at the end, first, as shown in FIG. 4, an arbitrary connection vector V is given to the PI at the start end.

Next, the steps 81 to S4 described above are performed for P2, and P
Through g, vector n, (string vector p, p,, ptp
Obtain the plane π whose normal vector is the cross product of s, and vI
Let the intersection of the extension Hti l and the plane π1 be the mouth.

Then, a quadratic Bezier curve with control points P, , Q, , P2 is generated, and it is converted to cubic to obtain control points Q, ′, Q,″ of the Bezier curve Ct.

Similarly, by giving a tangent vector v2 at the terminal end Pn and setting the intersection of its extension and the plane π7-2 (π2 in the example of FIG. 4) as a control point, the Bezier curve C* at the terminal end
-1 can be obtained.

The feature of this second method is that the Bezier curve extending from the passing point passes through the next point (for example, P9 to P8).
It is to be generated in such a way that it moves towards.

In other words, the control point u2 is on the plane parallel to the string vector ptp.
I'm looking for. Therefore, the larger the change in the distance between points, the more a different curve from the first method is obtained.

Using the above two methods, a boundary vAw4 such as a quadrilateral or trilateral is formed with a cubic Bezier curve while giving a passing point, and a model of a Bezier surface can be created by local modification of the 0 passing point. is easy; all you need to do is recalculate before and after the change. Furthermore, since the B-spline curve is not intermediately generated, it is also possible to generate a tangentially discontinuous curve as shown in FIG. 6, for example.

Note that when a boundary line network is formed using cubic Bezier curves using the method of the present invention to form a free-form surface model, even if each boundary line does not satisfy the tangent continuity condition, the tangent plane continuity condition In order to satisfy one of the requirements, one requirement is provided to connect all curved surfaces smoothly. One of the conditions for tangent plane continuity is that the normal vectors of the vector along the boundary of adjacent surface elements and the vector in the direction across the boundary point in the same direction with respect to both surface elements. A method of generating a curved surface with continuous tangent planes is disclosed in, for example, Japanese Patent Application No. 69385/1985 by the present applicant.

The generated three-dimensional free-form surface geometric model data is then input to a free-form surface cutting tool path generation system and converted into machining data for an NG milling machine (NC milling machine).

In the above description, the group of passing points is given in a three-dimensional space, but it is also possible to specify the group of passing points on a plane to generate a plane curve in Bezier representation. In this case, 2
Since the control points of the next and cubic Bezier curves can be drawn and obtained, the curves can be predicted intuitively.

Note that in the first method, as in the second method, the string vector P
, a line parallel to P3 is formed on P8, and this line and the plane π2
The intersection point with 02 may be set as 02.

〔Effect of the invention〕

According to the numerical processing method of the present invention, a cubic Bezier curve that passes through a given point group can be generated as a straight line without using a B-spline curve, so processing information can be generated efficiently and at high speed. Moreover, since local changes are extremely easy, it is possible to construct a highly interactive interface with the combiator, and shape modeling can be performed as intended by the designer.

[Brief explanation of the drawing]

FIG. 1 is a diagram showing a first method of generating a Bezier curve with designated passing points, which represents an embodiment of the present invention, FIG. 2 is a flowchart showing the steps of the first method, and FIG. 3 is a Bezier curve. FIG. 4 is a diagram showing the second generation method, FIG. 5 is a flowchart showing the steps of the second method, and FIG. 6 is a diagram showing the control points.
The figure is a diagram showing an example of a generated Bezier curve, Figure 7 is a diagram of a cross-sectional model of a workpiece, and Figure 8 is a diagram showing a conventional Bezier curve generation method that intermediately generates a B-spline curve. . In addition, in the symbols used in the drawings, P l"" P 4・−・−・−・−・・−・− Passing point π1~π3−−−・〜−−−−Plane 01~C++−1−・・－・・・－・・・It is a Bezier curve.

Claims (1)

[Claims] A method of numerically controlled machining along a curve passing through a given point group, in which four consecutive points P_1 to P_4 are taken out, and three of them P_2 to P_4 are processed from the center point P_3. The process of determining a plane π_2 that has a normal vector determined based on the information of the two chord vectors directed to the points on both sides and that passes through the point P_3, and the other three consecutive points P_1 to P_3 among the four points above. A process of determining a straight line l_2 that is on a plane passing through and passing through P_2, and determining an intersection Q_2 between the straight line l_2 and the plane π_2, and generating a quadratic Bezier curve with points P_2, Q_2, and P_3 as control points, This numerically controlled machining method consists of a process of converting this into a third order to generate a cubic Bezier curve, and obtaining numerically controlled machining data using the generated curve as the contour line of the workpiece.