JPH04287171A - Free curve preparing method - Google Patents

Abstract

PURPOSE:To generate a free curve similar to the sequence of dots by inputting this sequence of dots by setting a control side vector by using the connection of a third-order curve after approximating the sequence of dots to this third- order curve. CONSTITUTION:A third-order curve y=ax<3>+bx<2>+cx+d similar to dot sequence vectors q1-qn is generated by using a CAD/CAM (computer aided design/ computer aided manufacturing) method, for example. Afterwards, tangential vectors t1 and t2 of the third-order curve y=ax<3>+bx<2>+cx+d are detected and on the extending lines of these tangential vectors t1 and t2, control point vectors Pa1 and Pa2 are set. Therefore, a free curve R(t) similar to the dot sequence vectors q1-qn can be formed and the free curve can be generated by inputting the dot sequence vectors q1-qn. Namely, the free curve preparing method can be obtained to generate the Blzier curve of a shape connecting the inputted sequences of dots.

Description

[Detailed description of the invention]

0001

[Table of Contents] The present invention will be explained in the following order. Industrial application field Conventional technology (Fig. 17) Problem to be solved by the invention (Fig. 17) Means for solving the problem (Fig. 2) Effect (Fig. 2) Example (1) Overall CAD/CAM system Configuration (Figure 1) (2
) Creation of free curves (2-1) Generation of curve segments (Figures 1 to 9) (2-
2) Connection of curved segments (Figs. 10 to 16) (3) Effects of the embodiments (4) Effects of the invention of other embodiments

0002

[Industrial Field of Application] The present invention relates to a method for creating free curves, for example using CAD/CAM (computer aid
d design /computer aided
It can be applied to design devices using the manufacturing method.

0003

[Prior Art] For example, when designing the shape of an object with a free-form surface using a CAD method (geomet
ric modeling), a designer generally specifies multiple points (these are called nodes) in a three-dimensional space through which a curved surface should pass, and calculates a boundary curve network connecting the specified nodes using a desired vector function. This creates a curved surface expressed in a so-called wire frame. In this way, a large number of framework spaces surrounded by boundary curves can be formed (this process is called framework processing).

The boundary curve network formed by such framework processing itself represents the rough shape that the designer intends to design, and is created by a predetermined vector function using boundary curves surrounding each framework space. If it is possible to perform interpolation calculations on a curved surface that can be expressed, 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 the basic elements that make up the entire curved surface,
This is called patuchi.

By the way, in order to give the generated free-form surface a more natural outer shape as a whole, a patch is placed between two adjacent framework spaces across a shared boundary such that the condition of tangent plane continuity is satisfied at the shared boundary. A free curve creation method has been proposed in which the control edge vectors around the shared boundary are reset (Patent Application No. 60-2774).
No. 48).

As shown in FIG. 17, this free-form surface creation method uses patch vectors S(u
, v)1 and vector S(u,v)2 are expressed as a vector function vector S(u,v) consisting of a cubic Bezier equation, and 2
patch vector S(u,v)1 and vector S(u
, v)2, the node vector P(00) and vector P(3
0)1 , vector P(33)1 , vector P(03
), vector P(33)2, and vector P(30)2, the condition of tangent plane continuity holds at the shared boundary COM of adjacent patch vectors S(u,v)1 and vectors S(u,v)2. Set the control side vectors vector a1, vector a2, vector c1, vector c2 such that the control point vector P(11)1, vector P(12)1, vector P(11)2 , the principle is to reset the vector P(12)2.

If such a method is applied to other shared boundaries, the patch vector S (u, v) 1 and the vector S (u, v) 2 will eventually become smooth according to the condition of tangent plane continuity with the adjacent patch. can be connected to. here,
A vector function vector S(u, v
) is the following formula

[Equation 1] It is expressed using parameters u and v in the u direction and v direction, and shift operators E and F, and for the control point vector P(ij), the following equation

[Math 2]

[Math 3]

[Math 4]

[Math 5] have the following relationship.

Furthermore, a tangential plane means a plane formed by tangent vectors in the u direction and v direction at each point of the shared boundary. For example, for the shared boundary COM in FIG. 17, the patch vector S(u,v) 1 and vector S(u,
v) When two tangent planes are the same, the condition of tangent plane continuity holds true.

According to this method, the shape of the curved surface changes smoothly as a whole as intended by the designer.
Even object shapes that are difficult to design using conventional design methods can be easily designed.

0010

[Problem to be Solved by the Invention] By the way, we believe that it would be more convenient and easier to use the design device if the external shape of the object to be designed could be input as a sequence of points in such a design device. It will be done. To do this, it is necessary to create a wireframe model with a shape that connects the input points. Furthermore, it is necessary to express the free curves forming the wire frame model using Bezier curves.

The present invention has been made in consideration of the above points, and aims to propose a free curve creation method for creating a Bezier curve having a shape that connects a series of input points.

0012

[Means for Solving the Problem] In order to solve the problem, in the present invention, a control point vector Pa1, a control point vector Pa1,
Vector Pa2 is generated, and two node vectors Pa0 and vector Pa3 are generated based on the node vector Pa0, vector Pa3 and the control point vector Pa1, vector Pa2.
In a free curve creation method that generates a free curve R(t) expressed by a vector function connecting between Ivy cubic curve y=ax3
After generating +bx2 +cx+d, point sequence vector q1
Starting point vector q1 and ending point vector qn of ~qn
Correspondingly, a starting point vector q1a and an ending point vector qna are set on the cubic curve, and the cubic curve y=ax3 +b
Regarding the starting point vector q1a and the ending point vector qna on x2 +cx+d, the cubic curve y=ax3 +bx2
Detect the tangent vector t1 and vector t2 of +cx+d, set the start point vector q1a and end point vector qna on the cubic curve y=ax3 +bx2 +cx+d to the node vector Pa0 and vector Pa3, and set them on the extension line of the tangent vector t1 and vector t2. Control point vector Pa
1. By setting vector Pa2, the node vector Pa0, vector Pa3, and control point vector Pa1
, vector Pa2 is expressed as point sequence vector q1 ~ q
A free curve R(t) similar to n is generated.

[0013]

[Operation] After generating a cubic curve y=ax3 +bx2 +cx+d similar to the point sequence vector q1 to qn,
Detect the tangent vector t1 and vector t2 of the cubic curve y=ax3 +bx2 +cx+d, and use the tangent vector t
1. Control point vector Pa1 on the extension line of vector t2,
By setting the vector Pa2, you can easily obtain the point sequence vector q1.
A free curve R(t) similar to ~qn can be formed, and thereby a free curve can be generated by inputting the point sequence vector q1 ~qn.

[0014]

DESCRIPTION OF THE PREFERRED EMBODIMENTS An embodiment of the present invention will be described in detail below with reference to the drawings. (1) Overall configuration of CAD/CAM system In FIG. 1, 10 indicates the CAD/CAM system as a whole, and the free-form surface creation device 12 generates shape data DTS representing the free-form surface.
After creating the data, the tool path creation device 13 creates machining data DTCL for cutting.

That is, the free-form surface creation device 12 has a central processing unit (CPU), and by operating the input device 17 in response to the display on the display device 16, the free-form surface creation device 12 adds 3D to the wire frame model specified and input by the designer. After applying patches using the Bezier formula, shape data DTS of an object having a free-form surface is created by reconnecting the patches.

On the other hand, the tool path creation device 13 creates machining data DTCL for rough machining and finishing of the mold based on the shape data DTS, and then creates machining data DTCL for rough machining and finishing machining. , for example, to the NC milling machine 14 via a floppy disk 15. The NC milling machine 14 drives, for example, an NC milling machine based on the processing data DTCL,
As a result, a mold for the product represented by the shape data DTS is created.

(2) Creation of a free curve (2-1) Creation of a curve segment The free curve creation device 12 executes the processing procedure shown in FIG. Generate a group. That is, the free-form surface creation device 12 moves from step SP1 to step SP2, where the point sequence vector q1 input by the designer via the input device 17 is
, vector q2 , vector q3 , vector q4 ,
..., import the coordinate data of vector qm.

Next, the free-form surface creation device 12 moves to step SP3, and detects irregularities in the input point sequence. Here, if the point sequence cannot be represented by a cubic curve, the free-form surface creation device 12 divides the point sequence and each divided point sequence (
(hereinafter referred to as segments) can be expressed as cubic curves.

That is, as shown in FIG. 3, the point vector q
In the continuous point sequence from 1 to the point vector qm, there are two peaks, which indicates that the point sequence cannot be represented by a cubic curve. Therefore, in this case, the free-form surface creation device 12, as shown in FIG. The point sequence is divided into a second segment SGT2 up to the point vector qm that has passed the second peak and valley.

Next, the free-form surface generating device 12 moves to step SP4 and generates a cubic curve similar to the first segment SGT1 using the least squares approximation method. In other words, the cubic curve is

[Math 6] It can be expressed as

[0021] Therefore, this can be expressed as

[Formula 7] Now, let the x and y coordinate values of point vector qi be xi and yi, respectively, then the distance d(i) in the y direction between the cubic curve and point vector qi expressed by equation (7) is , the following equation

[Math. 8] It can be expressed as

Therefore, for the first segment SGT1, the distance d between each point vector q1, . . . , vector qn
The sum of squares S of (i) is calculated by the following formula

It can be expressed as [Equation 9], and the constants a and b are
, c, and d, it is possible to generate a cubic curve similar to the first segment SGT1. That is, (
9) Differentiate the equations with constants a, b, c, and d, and get the following equation:

[Math. 10]

[Math. 11]

[Math. 12]

[Math. 13] Since this can be expressed as a matrix, the following equation

It can be seen that it is sufficient to solve the simultaneous equations of [Equation 14].

Thus, the free-form surface creation device 12 executes the calculation process of equation (14) on the point group vectors q1, . . . , vector qn of the first segment SGT1,
As a result, as shown in FIG. 5, the point sequence vector q1,...
..., a cubic curve L1 similar to the vector qn is created.

Next, the free-form surface creation device 12 moves to step SP5, and corrects the coordinate data of the starting point vector q1 and the ending point vector qn for the first segment SGT1. Here, the free-form surface creation device 12 uses the following formula

[Number 1
5]

As shown in Equation 16, the starting point vector q1 and the ending point vector q
Substitute the x-coordinate values x1 and xn of n into equation (6).

As a result, as shown in FIG. 6, coordinate data (x1, y1a) and (xn, yna) on the cubic curve L1 corresponding to the x coordinate values x1 and xn are obtained, and the coordinate data (x1, y1a) is obtained. ) and (xn, yna
) are set as new starting points and ending points.

Next, the free-form surface creation device 12 moves to step SP6, and here, as shown in FIG. 7, the tangent vector of the cubic curve L1 at the new starting point vector q1a and ending point vector qna is detected. That is, the following equation

[Number 1
7] Since the cubic curve L1 can be differentiated to express its slope, coordinate data x1 and xn can be expressed in equation (17).
By substituting , it is possible to detect the slope of the cubic curve L1 at the starting point vector q1a and the ending point vector qna.

Therefore, the tangent vector t1 of the starting point vector q1a can be expressed as follows:

[Math. 18] If expressed as, on the starting point vector q1a, the following formula

By establishing the relational expression [Equation 19], the free-form surface creation device 12
is the following formula

The tangent vector t1 of the starting point q1a is detected by performing the calculation process as follows.

Similarly, the tangent vector t2 of the end point vector qna can be expressed as

[Math. 21] If expressed as, on the end point vector qna, the following formula

By establishing the relational expression [Equation 22], the free-form surface creation device 12
is the following formula

The tangent vector t2 of the end point vector qna is detected by performing the calculation process as follows.

As a result, the free-form surface generating device 12 detects the tangent vector t1 and vector t2 necessary for generating a Bezier curve. Next, free-form surface creation device 1
Step 2 moves to step SP7, where a Bezier curve similar to a point sequence is generated using the coordinate data of the tangent vector t1 and vector t2, the start point vector q1a, and the end point vector qna.

Here, as shown in FIG. 8, the Bezier curve is
Using the third-order Bezier equation, the following equation,

[Number 2
4] A parametric spatial curve vector R(t) expressed as
It is expressed as

Here, t is one of the nodal vectors P0
to the other node vector P3 in the direction along the curve segment KSG, the following equation

This is a parameter that changes from a value of 0 to a value of 1 as expressed by the following equation. In this way, the curve segment KSG expressed by the cubic Bezier equation can be created by specifying the two control point vectors P1 and P2 between the node vector P0 and vector P3 using the shift operator E. Each point on segment KSG is expressed by the following formula

It is expressed as a position vector R(t) from the origin O of the xyz space by expanding .

Here, the shift operator E is calculated using the following equation for the control point vector Pi on the curve segment KSG.

[Number 27
]

[Math. 28] have the following relationship.

Therefore, by expanding equation (24) and substituting the relationship in equation (27), we get the following equation:

It can be calculated as follows, and as a result, equation (26) is obtained.

Thus, each curve segment KSG1, KSG2, KSG3 represented by the Bezier curve is (26)
Two node and control point vectors P, respectively, based on Eq.
(0)1-3, vector P(1)1-3, vector P(2)1-3, and vector P(3)1-3, and the node vector P(0)1-3 and P
(3) By setting control point vectors P(1)1 to 3 and vectors P(2)1 to 3 between 1 and 1, control is performed through node vectors P(0)1 to 3 and P(3)1. Point vector P(0)1-3, vector P(1)1-3,
vector P(2)1-3 and vector P(3)1-3
It can be set to a shape determined by .

Based on this setting principle, the free-form surface creation device 12 sets the starting point vector q1a and the ending point vector qna to the node vector P0 and vector P3 of the curve segment KG, and then sets the control point vector P1 and vector P2 are temporarily set. That is, the free-form surface creation device 12 sets the initial values of the values α and β, and calculates the following equation

[Math. 30]

[Math. 31]

[Math. 32]

##EQU00003## Then, a Bezier curve determined by the control point vector P1 and vector P2 expressed by equations (31) and (32) is provisionally set.

Next, the free-form surface creation device 12 calculates the starting point vector q1 for the cubic curve generated in step SP5.
Divide equally between a and the end point vector qna, and place 1 on the curve
One-point division point vector PS1, ..., PSi, ..., P
Generate S11. Furthermore, the free-form surface creation device 12
Each division point vector PS1, ..., PSi, ..., PS1
The control point vector P1 and vector P2 are reset so that the distance from 1 to the temporarily set Bezier curve becomes smaller, thereby generating a Bezier curve similar to the point sequence vector q1, . . . , vector qn.

In other words, the free-form surface creation device 12 has the following formula:

[
A Bezier curve vector R(ti) is generated so that the function f expressed by the following equation is minimized.

Here, PSix and PSiy represent the x and y coordinate values of the dividing point PSi, respectively, and R(ti)x
, R(ti)y is the point vector R(ti) on the Bezier curve vector R(t) closest to each dividing point PSi.
represents the x and y coordinate values of Here, in equation (34), if we differentiate by the values α and β and set the value to 0, we can obtain the following equation

[Math. 35]

By performing Taylor expansion and linearizing this, the values α and β can be selected by applying the Newton-Raphson method.

That is, by transforming equations (35) and (36), we obtain the following equation:

[Math. 37]

[Math. 38] Since it can be expressed as

Express it as a determinant as shown in Equation 39 and solve it using the Newton-Raphson method.

[0040] As a result, the free-form surface creation device 12 calculates the value α
After selecting and β, substitute them into equations (31) and (32), and thereby control point vector P1a and vector P2a
Set. The point sequence vector q1 thus input ~
If the vector qm can be represented by one cubic curve, it is possible to generate a Bezier curve vector R(t) similar to the point sequence vector q1 to qm.

On the other hand, the input point sequence vector q
1 to vector qm cannot be represented by one cubic curve, the free-form surface creation device 12, in step SP8, calculates the Bezier curve vector R(t
) is generated, a negative result is obtained and the process returns to step SP4. As a result, the free-form surface generating device 12 generates the Bezier curve vector R(t) for the following segment, and then proceeds to step SP9 and ends the processing procedure.

(2-2) Connecting curve segments When the above processing procedure is executed to express a point sequence with multiple curve segments, the curves may be unnaturally connected or the curve segments may be There is a risk that they will leave. For this reason, the free-form surface creation device 12 continues as shown in FIG.
Perform processing steps to connect curve segments.

That is, from step SP20 to step S
Proceeding to P21, the free-form surface creation device 12 sequentially changes the parameter t for each of the adjacent curve segments A and B and detects the values, thereby creating a curve on the curve segments A and B as shown in FIG. 1 each
One-point division point vector qa1, vector qa2,...
Vector qa11, vector qb1, vector qb2
, . . ., generates a vector qb11.

Next, the free-form surface creation device 12 moves to step SP22, where it judges whether the end point vector qa11 and the start point vector qb1 of curve segments A and B match, and if a negative result is obtained here, Reset the start point and end point.

That is, as shown in FIG. 12, if the end point vector qa11 and the starting point vector qb1 do not match, the following equation

[Math. 40]

[Math. 41]

[Math. 42]

[Formula 43] is executed, and the midpoint between the end point vector qa11 and the start point vector qb1 is determined as the node vector Pa3n of the curve segment A, the end point vector qa11 and the node vector Pb0n of the curve segment B, and the start point vector qb.
Set to 1. Next, the free-form surface creation device 12 moves to step SP23, where the following equations are used for the curve segments A and B, respectively.

[Math. 44]

45 is executed to detect the vector av from the node vector Pa3n to the control point vector Pa2 and the vector bv from the node vector Pb0n to the control point vector Pb1.

Furthermore, as shown in FIG. 13, the free-form surface creation device 12 calculates the following equation for vector av and vector bv.

A cross product vector cv is obtained as shown in Equation 46, and the calculation process of the following equation is performed on the cross product vector cv.

[Math. 47]

[0047] As a result, the free-form surface creation device 12 reduces the angle formed by the vector av and the vector bv to 1.
For the vector cv to be divided into /2, a vector dv orthogonal to the vector cv is generated, and the vector dv is divided into curve segment A.
and B's control edge vector.

Next, the free-form surface creation device 12 moves to step SP24, where the Bezier curves for the curve segments A and B are re-formed. That is, the free-form surface creation device 12 calculates the nodal vector P for the curve segment A.
Let the unit vector from a0 to the adjacent division point qa1 be vector vector t1, and the unit vector of vector dv be vector vector t2.

[Math. 48]

[Math. 49]

[Number 50]

[Math. 51] Put it as.

Furthermore, the free-form surface creation device 12 (34)
As described above for equation (39), the Newton-Raphson method is used to select the values α and β, thereby resetting the curve segment A as shown in FIG.

Next, the free-form surface creation device 12 similarly resets the control side vector for the curve segment B, connects the curve segments A and B so that the tangent changes smoothly, and then proceeds to step SP25. Move,
The processing procedure ends.

[0051] In this way, even in cases where it is difficult to represent with a single cubic curve, a free curve similar to the point sequence input by the operator can be formed using multiple curve segments, and the corresponding free curve The usability of the creation device 12 can be improved. In fact, according to the results of experiments performed by inputting the point sequence q1 to q16, as shown in FIGS. 15 and 16, even when complex changes occur, the Bezier curve can be smoothed after being generated I was able to confirm that I could connect.

In this case, for example, if the connection process is repeated sequentially from the left to the right curve segments, the error in the last curve segment will increase. For this reason, in this embodiment, curves and segments are successively connected in the order specified by the user.

(3) Effects of Embodiment According to the above configuration, after approximating a point sequence to a cubic curve, a Bezier curve is generated based on the tangents of the starting point and the contact point, so that the point sequence can be approximated. A similar free curve can be generated, and a free curve can be generated by inputting a sequence of points. Therefore, the usability of the free-form surface creation device can be improved accordingly.

(4) Other Embodiments In the above embodiment, 11 division points are generated on the cubic curve, and the control side vector is calculated using the least squares method using the coordinate data of the division points. Although the case where the number of divisions is set has been described, the present invention is not limited to this, and the number of divisions can be freely set as necessary.

Further, in the above embodiment, the case where a wire frame model is generated has been described, but the present invention is not limited to this, but can be widely applied to cases where free curves are created as necessary.

[0056]

Effects of the Invention As described above, according to the present invention, after approximating a point sequence to a cubic curve, by setting a control side vector using the tangent to the cubic curve, a free resemblance to the point sequence can be obtained. It is possible to obtain a free curve creation method that can generate a curve, and can thus input a sequence of points and generate a free curve similar to the sequence of points.

[Brief explanation of the drawing]

FIG. 1 is a block diagram showing the overall configuration of a CAD/CAM system according to an embodiment of the present invention.

FIG. 2 is a flowchart showing the procedure for generating the curve segment.

FIG. 3 is a schematic diagram showing an input point sequence.

FIG. 4 is a schematic diagram for explaining division of a point sequence.

FIG. 5 is a schematic diagram for explaining generation of a cubic curve.

FIG. 6 is a schematic diagram illustrating a start point and end point resetting process.

FIG. 7 is a schematic diagram for explaining tilt detection.

FIG. 8 is a schematic diagram for explaining a Bezier curve.

FIG. 9 is a schematic diagram for explaining the setting of a Bezier curve.

FIG. 10 is a flowchart showing a curve segment connection process.

FIG. 11 is a schematic diagram showing curved segments before connection.

FIG. 12 is a schematic diagram for explaining control point resetting processing.

FIG. 13 is a schematic diagram illustrating a control side vector setting process.

FIG. 14 is a schematic diagram showing connected curve segments.

FIG. 15 is a schematic diagram showing an actually inputted point sequence.

FIG. 16 is a schematic diagram showing an actually generated free curve.

FIG. 17 is a schematic diagram for explaining a free-form surface.

[Explanation of symbols]

10...CAD/CAM system, 12...Free-form surface creation device, P(11), P(11)1, P(11)2
...Node, control point, R(t) ...Free curve, S(u,
v), S(u,v)1...patch.

Claims (1)

[Claims] Claim 1: A free curve that generates a control point between two nodes in a three-dimensional space, and generates a free curve expressed by a vector function connecting the two nodes based on the node and the control point. In the creation method, a cubic curve similar to the point sequence is generated based on the coordinate data of the input point sequence, and then a starting point and a starting point and an end point are created on the cubic curve, corresponding to the starting and ending points of the point sequence. Set an end point, detect a tangent to the cubic curve for the starting point and ending point on the cubic curve, set the starting point and ending point on the cubic curve to the node, and set the control point on an extension of the tangent. By setting
A method for creating a free curve, comprising generating the free curve similar to the point sequence represented by the nodes and control points.