JP3408866B2 - Method and apparatus for generating draft surface between curve and curved surface - Google Patents

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for generating a draft surface between a curved surface and a curved surface and its apparatus, and more particularly to a shape deformation operation method in a three-dimensional shape model processing apparatus.
For example, it is applied to a draft surface generation method for a three-dimensional shape model including a curve and a curved surface.

[0002]

2. Description of the Related Art In the conventional draft plane generation method, for example, the following methods have been proposed. Regarding the method for use on a two-dimensional plane, see, for example, JP-A-5-8.
No. 9210. In this publication,
CAD (Computer Aided Design) for designing molds to be used for plastic molding by deforming the shape to be processed
The present invention relates to a CAM (Computer Aided Manufacturing) device. In a draft line transformation in which a line to be drafted is rotated and moved by a specified angle, not only the line segment but also the conventional one among the shapes in which the end points exist on the line to draft The arcs that have not been deformed are also deformed so as to maintain the relationship of continuity, etc. according to the deformation of the draft line, and the arcs that have endpoints on the line to be drafted and are not in contact are draft lines. You can specify either parallel movement mode or intersection cut mode when deforming following the deformation of
Information necessary for deformation when changing the draft angle is added to the line to be drafted.

Further, in the case of generating a draft of a triangular element, a method limited to the case where the draft angle is not explicitly given is disclosed in, for example, JP-A-5-29792.
No. 1 publication. In this publication, the shape of the SM (surface model) is formed in order to easily obtain the shape data of the draft surface of the die, which makes it easier to pull out the die and reduces the defects in the shape of the product formed by the die. Data and
The shape data of a parting line of a reference metal mold passing through a position separated from the SM is stored in a computer, and the SM of the parting line of the reference metal mold is stored.
Draw a tangent line with a draft angle to the shape of, and find a point that is apart from that contact point by a predetermined length along the tangent line toward the dividing line of the reference metal mold, and use these two points as the reference metal. One round is made around the parting line of the mold to automatically obtain the draft surface of the mold.

[0004]

However, in the above case, the method of expressing the draft surface shape in the three-dimensional space is not described. Therefore, there are problems such as not being able to confirm the shape while displaying the shape of the designed draft surface in the three-dimensional space on the screen, and being unable to exchange data with the three-dimensional model processing device. It was happening. Also, in the above case, the draft angle is not explicitly given, and a mathematical approach is explicitly shown for the means for generating the intersection line between the draft surface and the curved surface and the approximate expression means for the draft surface. It has not been. Therefore, when the draft angle is explicitly specified, the draft surface cannot be generated, it cannot be guaranteed whether or not the draft surface can be created. Even if the draft surface can be created, the created curved surface is There have been problems such as not being able to guarantee that data can be exchanged with the three-dimensional model processing device, and not being able to guarantee that accuracy can be improved.

The present invention has been made in view of the above circumstances, and when a draft surface is generated between a curved line and a curved surface in a three-dimensional space by using a three-dimensional shape model processing device, Generates a three-dimensional draft surface when the draft angle is 0 degrees and the draft angle at any point on the curve is explicitly specified. Data can be exchanged as the draft surface expression. An object of the present invention is to provide a method and apparatus for generating a draft surface between a curved surface and a curved surface so that a simple curved surface expression can be used.

[0006]

In order to solve the above-mentioned problems, the present invention (1) obtains the intersection of a straight line on a draft surface passing through a point on a curved surface and the curved surface, and stores it in a storage device. Then, the movement vector of the intersection of the draft surface and the curved surface when the draft surface follows the curve is obtained, and the B- spline composed of C ¹ continuous curve segments is formed from the intersection and the movement vector. Generate a curve, perform linear interpolation between the curve that the draft surface passes and the generated B-spline curve,
Generating a B- spline patch or a rational B- spline patch, or (2) in a three-dimensional shape model processing apparatus that handles curved surfaces, includes a direction with a draft angle of 0 degrees and a curve with a draft angle specified. An intersection line generation means for generating an intersection line between a draft surface and a curved surface, a curve with a draft angle specified, and an intersection line generated by the intersection line generation means are used to approximate the draft surface. It is characterized in that it comprises a B- spline patch for expressing or a patch generating means for generating a rational B- spline patch.

[0007]

[Function] An intersection of a straight line on a draft surface passing through a point on a curve and a curved surface is obtained and stored in a storage device, and then the draft surface and the curved surface when the draft surface follows the curve. Find the movement vector of the intersection with and. B-consisting of C ¹ continuous curve segments from the intersection and its movement vector
A spline curve is generated, linear interpolation is performed between the curve that the draft plane passes and the generated B-spline curve, and a B-spline patch or a rational B-spline patch is generated. In this way, when the draft angle on the curve and the direction of the draft angle of 0 degrees are explicitly designated, it is possible to generate a draft surface between the curve and the curved surface.

[0008]

Embodiments will be described below with reference to the drawings. 1 to 5 are configuration diagrams for explaining an embodiment of a draft surface generation method between a curve and a curved surface according to the present invention, in which C _a is a curve, _Sa is a draft surface, S _b is a curved surface intersecting with the draft surface, Q _i is a part of the curve C _a , L _i is a straight line on the draft surface at one point on the curve, P _i is a straight line L _i on the draft surface and the curved surface S. intersection with the _b, C _b is draft surface S _a
And the curved surface S _b , C _bj is the i-th segment of the curve C _b , and S _c is a B-spline patch or a rational B that approximates a draft surface by linearly interpolating between C _a and C _b. -It is a spline patch.

The present invention is for generating a draft surface between a curved line and a curved surface in a three-dimensional shape model processing apparatus. This draft surface generation method includes the following four processes from step S1 to step S4. Here, an outline of four processes will be described using a process of generating a draft surface between the curve C _a and the curved surface S _b shown in FIG. 1. The outline of this process is shown in the flowchart of FIG.

In the following description, the curve C _a is a polynomial curve or rational curve defined by the parameter u (0 ≦ u ≦ 1). The draft angle A is expressed as a function A (u) defined by the parameter u, and the present invention can be used to generate a gradually changing draft surface. The unit vector R representing the direction of the draft angle of 90 degrees is defined by the parameter u and is represented by the following equation.

[0011]

[Equation 1]

The draft surface S _a has parameters u, v (0 ≦ u
≦ 1,0 ≦ v ≦ ∞), and is expressed by the following equation.

[0013]

[Equation 2]

The curved surface S _b has parameters u and v (0 ≦ u, v ≦
It is a parametric surface defined by 1). The draft line L _i is a linear parametric curve defined by the parameter v. The draft line L _i has the parameter u _i =
When passing _a point Q _i = C _a (u _i ) on the curve C _{a at} i / nm (i = 0, 1, 2, ..., nm-1, nm), it is expressed by the following formula.

[0015]

[Equation 3]

A curved surface S _c approximating a draft surface is a B-spli defined by parameters u and v (0≤u, v≤1).
It is ne patch or rational B-spline patch. The number of this curved surface S _c finally generated is expressed as 1.
It should be noted that steps S1 to S3 describe the intersection line generation means, and step S4 describes the B-spline patch or rational B-spline patch generation means.

Step S1; First, the curve C _a is equally divided into nm lines in the space of the parameter u, and nm + 1 division points Q are obtained.
The draft line L _i at _i (i = 0, 1, 2, ..., nm-1, nm) is calculated. Then, these draft lines L _i
And the intersection point P _i between the curved surface S _b and the curved surface S _b is obtained and stored in the storage device (FIG. 2).

Step S2: A movement vector V _i indicating how the intersection P _i moves when the draft line L _i calculated in step S1 is about to slide on the curve C _a.
Is calculated and stored in the storage device (FIG. 3). However, among the intersection points P _i , i (i = 0, m, 2m, ..., (n
-1) The processing shown here is performed only for m, nm).

Step S3: Intersection P of step S1
_i and the movement vector V _i in step S2,
n curve segments C connected to C ¹ continuous on _b
B-spli composed of _bj (j = 0, 1, 2, ..., N-1)
The ne curve C _b is generated and stored in the storage device (FIG. 4). 1
The curve segment C _{bj of the} book has two intersections P _jm and P _{(j + 1) m.}
Is an end point, and the first-order differential vector with respect to u of the curve segment C _bj faces the same direction as the movement vectors V _jm and V _{(j + 1) m at} the intersections at P _jm and P _{(j + 1) m} , respectively. Is generated. The B-spline curve C _b approximately represents the locus drawn by the intersection of the draft surface S _a and the curved surface S _b .

Step S4: Curve C _a and the step S
The loft surface S _c ′ is generated by linearly interpolating between the curve C _b stored in 3 and the curved surface C _b, and this curved surface is converted into a B-spline patch or a rational B-spline patch S _c (FIG. 5). And
After the curved surface S _c after the conversion is stored in the storage device, the process ends.

The curved surface S _c has its boundary defined by _a curve C _a defining a draft angle and an intersection line C _b between the draft surface and the curved surface S _b . In addition, n + 1 parameters u _i =
i / nm (i = 0, m, 2m, ..., (n-1) m, nm)
In each of the parameters of the straight line S _c on the curved surface S _c
(u _i , v) and the straight line L _i = S _a (u _i , v) on the draft surface S _a
Are on the same straight line. Therefore, the generated surface S _c
Is a curved surface that efficiently approximates the original draft surface S _a . It is also possible to improve the approximation accuracy of the draft surface by increasing the number of generated curve segments C _bj .

7 to 10 are views for explaining a method of generating an intersection of a draft surface and a curved surface. In the drawings, Q _a is a point on the curve C _a , and N is a draft angle of 0 °. A unit vector representing a direction, T is _a tangent vector at one point on the curve C _a , R
Unit vector representing the direction of the draft angle of 90 degrees, A is draft angle, L _a is a draft line passing through the point Q _a, the intersection between P _a 'is a draft line L _a polyhedral approximation songs S _b , P _a is the intersection of the draft line L _a and the curved surface S _b , P _b is the point on the curved surface S _b , F
_b is a tangent plane of the curved surface S _b, .differential.S _b / ∂u the surface S first derivative vector for u at a point on the _b, .differential.S _b / ∂v one derivative vectors for v at a point on the curved surface S _b, .delta.u _b and .delta.v _b are real numbers, and other portions of the same functions as in FIG. 1 to FIG. 5 are denoted by the same reference numerals.

Here, the method for obtaining the intersection of the draft surface and the curved surface shown in step S1 will be described in detail. According to the method of the present invention, the parameter u _i of the curve C _a is first determined, and the draft line L _a is arranged so as to pass on the curve C _a in the parameter and have a draft angle of A. Therefore, the intersection point P _a between the draft surface S _a and the curved surface S _b is a function of the parameter u.

The method of obtaining the intersection point P _a from the parameter u is realized by the following steps S11 to S19. FIG. 11 shows a flow chart of a method of generating an intersection of a draft line and a curved surface according to the present invention. This method
The geometric Newton method is used. That is, the initial point of the intersection is set on the draft and the curved surface, and these points are converged to the correct intersection.

[0025] Step S11; First, a point Q _a on the curve C _a in the parameter u, tangent vector T at that point, determine the direction R of the draft angle of 90 degrees from the direction N of draft angle of 0 degrees, the storage device To store. Then, the draft angle A for the parameter u is obtained and stored in the storage device (FIG. 7). Step S12; point Q _a on the curve C _a in the parameter u, the direction N of the draft angle of 0 degrees at that point, 90 degrees direction R, from the draft angle A, determine the draft line L _a, the storage device To store. Step S13: Calculate an intersection point P _a ′ between the curved surface S _b approximated to the polyhedron and the draft line La, and set _a point P _b on S _b closest to this point as an initial point P _b of the geometric Newton method. , In the storage device (FIG. 8). Further, the parameters (u _b , v _b ) of the curved surface S _b corresponding to the point P _b are calculated and stored in the storage device. Step S14; tangent plane F of the curved surface S _{b at} the starting point P _b
Calculate _b and store in storage. The normal vector n of the tangent plane of the parametric surface S is the parameter (u, v)
When is given, it can be calculated from the following formula.

[0026]

[Equation 4]

Step S15: The intersection point P _a between the draft line L _a obtained in step S12 and the tangent plane F _b obtained in step S14 is calculated and stored in the storage device (FIG. 9). Step S16; If it can be considered that P _a and P _b are the same point, that point becomes the intersection of the draft surface S _a and the curved surface S _b . That is, if the threshold value of the geometric Newton method is ε,

[0028]

[Equation 5]

When the above is true, P _a can be regarded as the intersection of the draft surface S _a and the curved surface S _b . In this case, the process proceeds to step S19 described later. Step S17: Using P _a , the next start parameter (u _b ′, v _b ′) on the curved surface S _b is calculated and stored in the storage device. The following equation is used to obtain the starting parameter (u _b ′, v _b ′).

[0030]

[Equation 6]

To calculate δu _b , δv _b , P _a −P _b
(Fig. 10) and two vectors

[0032]

[Equation 7]

The inner product of and is taken to obtain a binary linear equation relating to δu _b and δv _b . Δ obtained by solving this equation
new starting parameter u _b ′ = u _b + δu from u _b , δ v _b
Calculate _b , v _b ′ = v _b + δv _b . Step S18: Substitute the newly obtained start parameter (u _b ′, v _b ′) into (u _b , v _b ). A point P _b on the curved surface is newly calculated from the parameters of this. P
After all (u _b , v _b ) are stored in the storage device, the process returns to step S14. Step S19: The intersection point P _a is stored in the storage device, and all processing is terminated.

In step S1, the curve C _a is equally divided into nm lines in the parameter space, and draft lines are arranged on all the division points. Therefore, the parameter u of the curve C _a is set to u = i / nm (i = 0, 1, 2, ..., nm-1, nm)
Then, the intersection points P _i corresponding to all _i are calculated and stored in the storage device. 12 to 14 are diagrams for explaining a method of generating a movement vector of an intersection. N in the figure
_a is _a normal vector at one point on the curved surface S _a , n _b is a normal vector at one point on the curved surface S _b ,
Portions having the same functions as those in FIGS. 5 and 7 to 10 are designated by the same reference numerals.

[0035] The step S1, when the draft line L _i is to let slip the upper curve C _a, and requests the motion vector indicating the motion how the intersection of the draft line L _i and the curved surface S _b . Here, the method of generating the movement vector will be described in detail. At the intersection point P _i , the draft surface S _a
Is a tangent plane of F _a , and _a normal vector of the tangent plane F _a is n _a ,
Curved S _b on the tangent plane F _b, tangent plane F when _b of the normal vector to the n _b normal vector n _a, n _b is expressed by the following equation (12, 13).

[0036]

[Equation 8]

Using n _a and n _b obtained from the above equation,
The movement vector V _i is expressed by the following equation (FIG. 14).

[0038]

[Equation 9]

[0039] Thus, the movement vector V _i when the parameters of the draft plane _{S a (u a, v a} ), the curved surface S _b parameters (u _b, v _b) is determined,

[0040]

[Equation 10]

It is obtained by calculating In step S2, the parameter u of the curve C _a is set to u = i / nm (i
= 0, m, 2m, ..., nm), and n + 1 movement vectors V _i corresponding to _i are calculated and stored in the storage device.
This processing is performed only for i that is a multiple of m, because the movement vector corresponding to i other than the number of stories of m is not needed when the curve representing the locus of the intersection is generated in step 3. .

15 to 17 are explanatory views for generating a curve representing a locus. The step S3 is a part for generating a single curve that approximately expresses the locus of the intersection of the draft surface and the curved surface. Here, the intersection point P _i (i = 0, 1,
2, ..., nm-1, nm) and its movement vector V _i (i =
0, m, 2m, ..., (n-1) m, nm) will be described in detail. According to the method of the present invention, the continuous nm + 1 intersection points P _i and the n + 1 movement vectors V _i are composed of C ¹ continuous n curve segments C _bj (j = 0, 1, ..., N−1). One B-
Generate a spline curve (FIGS. 15 and 16).

The method of generating the B-spline curve C _b will be described below. First, nm + 1 consecutive intersection points P _i (i
= 0, 1, 2, ..., nm-1, nm), n + 1 movement vectors V _i (i = 0, m, 2m, ..., (n-1) m, n
m) is taken out from the storage device. Using these, a B-spline curve C _b satisfying the following conditions is generated.

Condition; n-1 parameters u _i = 1 /
nm (i = m, 2m, ..., (n-1) m) in the curve C _b is C ¹ continuous. Condition: The curve C _b has n + 1 intersections P _i (i = 0, m, 2
m, ..., (n−1) m, nm). Condition; n + 1 parameters u _i = i / nm (i = 0,
m, 2m, ..., (n-1) m, nm), the curve C
The first derivative vector at the point on _b points in the same direction as V _i . Condition; n + 1 parameters u _i = i / nm (i = 0,
m, 2m, ..., (n−1) m, nm), the generated point on the curve C _b coincides with P _i . Condition; n + 1 points P _i (i = 0, m, 2) on the curve C _b
m, ..., the magnitude of the first derivative vector at (n-1) m, nm ) , the following n (m-1 represented by the formula) point on the curve C _b in the number of parameters u _{jm + k} and The sum E of the square of the distance from the corresponding intersection P _{jm + k} is determined so as to be the minimum.

[0045]

[Equation 11]

To satisfy the conditions, the B-spline curve may be determined as expressed by the following equation.

[0047]

[Equation 12]

Where K is the knot vector N _i , ₄ (u)
Is the B-spline basis function, and D _i is the control point of the B-spline curve. Conditions and n + to meet the conditions
One intersection point P _i (i = 0, m, 2m, ..., (n-1) m,
nm), and n + 1 movement vectors V _i and 2n unknowns t _2j and t _{2j + 1} (j = representing the magnitude of the movement vector).
0, 1, 2, ..., N-2, n-1) may be used to determine the control points D _i so that the relationship expressed by the following equation is established.

[0049]

[Equation 13]

To satisfy the condition, t _2j and t _{2j + 1} may be determined so that the relational expression expressed below holds.

[0051]

[Equation 14]

Substituting this equation into equation (2), B-splin
The e curve C _b has n + 1 unknowns t _j (j = 0, 1, 2,
, N-1 and t (2n-1). When E expressed by Expression (1) is once differentiated for each of these n + 1 unknowns t _j , n + 1 n + 1 linear equations regarding t _j expressed by the following expressions are obtained.

[0053]

[Equation 15]

To satisfy the condition, t _j may be determined so as to satisfy these equations. That is, by solving the n + 1 n + 1 element linear equations related to t _j expressed by the above equation (2), t _j satisfying the condition is determined. From the above, B-spline that meets the above five conditions
The curve C _b can be calculated (FIG. 17). In step S4, the curve C _b and the curve C _a stored in step S3 are linearly interpolated to generate a loft surface S _c ′ represented by the following expression, and this curved surface is subjected to a B-spline patch or rational B. -spline Convert to a curved surface represented by S _c in the patch.

[0055]

[Equation 16]

Here, when the curve C _a is a polynomial curve, the curve C _a can be represented by _a B-spline curve, so the loft surface S _c ′ can be converted into a B-spline curved surface. When the curve C _a is a rational curve, the curve C _a can be represented by _a rational B-spline curve, so that the loft surface S _c ′ can be converted into a rational B-spline patch. Therefore, when the curve C _a is represented by _a B-spline curve, the converted surface S _c is a B-spline.
Express in a patch, otherwise rational B-spline
Express in patches. Then, the converted curved surface S _c is stored in the storage device, and the process ends.

As described above, if the present invention is introduced and a draft line is generated by sliding a draft line on a curved line at a designated draft angle so as to intersect a curved surface, a draft surface can be obtained. Can be solved. Also, B-spline patch or rational B
The second problem can be solved by approximating the draft surface using the -spline patch.

[0058]

As is apparent from the above description, according to the present invention, it is possible to generate a draft surface between one curve and one curved surface in a three-dimensional shape model processing apparatus including curved surfaces. This draft surface S _c has a parameter u =
i / nm, (i = 0, m, 2m, (n-2) m, (n-
1) In m), it lies on the exact draft plane expressed by equation (1). As a curved surface that expresses a draft surface, B-s
It uses pline patch or rational B-spline patch. Therefore, it becomes easy to handle the draft surface with the three-dimensional processing apparatus.

[Brief description of drawings]

FIG. 1 is a diagram (No. 1) for explaining an embodiment of a draft surface generation method between a curved line and a curved face according to the present invention.

FIG. 2 is a diagram (No. 2) for explaining one embodiment of the draft surface generation method between a curved surface and a curved surface according to the present invention.

FIG. 3 is a diagram (No. 3) for explaining an embodiment of a draft surface generation method between a curved surface and a curved surface according to the present invention.

FIG. 4 is a diagram (No. 4) for explaining an example of a draft surface generation method between a curved line and a curved surface according to the present invention.

FIG. 5 is a diagram (No. 5) for explaining one embodiment of the draft surface generation method between a curved surface and a curved surface according to the present invention.

FIG. 6 is a flowchart of a draft surface generation method according to the present invention.

FIG. 7 is a diagram (No. 1) for explaining a method of generating an intersection of a draft surface and a curved surface according to the present invention.

FIG. 8 is a diagram (No. 2) for explaining a method of generating an intersection of a draft surface and a curved surface according to the present invention.

FIG. 9 is a diagram (No. 3) for explaining a method of generating an intersection of a draft surface and a curved surface according to the present invention.

FIG. 10 is a diagram (No. 4) for explaining a method of generating an intersection of a draft surface and a curved surface according to the present invention.

FIG. 11 is a flowchart of a method for calculating an intersection line between a sloped surface and a curved surface according to the present invention.

FIG. 12 is a diagram (No. 1) for explaining a method of generating a movement vector of an intersection according to the present invention.

FIG. 13 is a diagram (No. 2) for explaining a method of generating a movement vector of an intersection according to the present invention.

FIG. 14 is a diagram (No. 3) for explaining a method of generating a movement vector of an intersection according to the present invention.

FIG. 15 is an explanatory diagram (No. 1) for generating a curve representing a locus according to the present invention.

FIG. 16 is an explanatory diagram (No. 2) for generating a curve representing a locus according to the present invention.

FIG. 17 is an explanatory diagram (No. 3) for generating a curve representing a locus according to the present invention.

[Explanation of symbols]

C _a ... Curve, S _a ... Draft surface, S _b ... Curved surface intersecting with draft surface, Q _i ... Part of curve C _a , L _i ... Straight line on draft surface at one point on curve, P _i ... intersection of straight line L _i and curved surface S _b on draft surface, C _b ... draft surface S _a and curved surface S _b
, C _bj ... i-th segment of curve C _b , S _c ...
A B-spline patch or a rational B-spline patch that linearly interpolates between C _a and C _b and approximately expresses a draft surface.

Claims (2)

(57) [Claims] 1. A draft surface when a straight line on a draft surface passing through a point on a curved surface and an intersection point of the curved surface are obtained and stored in a storage device, and then the draft surface follows a curved line. And a curved surface are generated, and a B- spline curve composed of C ¹ continuous curve segments is generated from the intersecting point and the movement vector, and a curve through which the draft surface passes and the generated B- spline. Linear interpolation between curves, B- spli
A method for generating a draft surface between a curved surface and a curved surface, characterized by generating an ne patch or a rational B- spline patch. 2. A three-dimensional shape model processing apparatus for handling a curved surface, which generates an intersection line for generating an intersection line between a draft surface and a curved surface including a curve having a draft angle of 0 degree and a draft angle specified. B- spline patch or rational B- spli that approximately expresses the draft surface using the means, the curve with the draft angle specified, and the intersecting line generated by the intersecting line generating means.
A draft surface generating device between a curved surface and a curved surface, which comprises a patch generating means for generating an ne patch.