JP5458440B2 - Curved surface generation device and curved surface generation program - Google Patents

Description

  The present invention relates to a curved surface interpolation technique for applying a curved surface to an N-side area of a three-dimensional model surface, and more particularly to a curved surface generation apparatus and a curved surface generation program for generating a curved surface that fits an N-side area including a concave shape.

  In a shape deformation operation in a three-dimensional CAD system, a trimmed curved surface is often generated. The boundary ridgeline that trims the curved surface is a curve that is on the curved surface within an allowable error range. Therefore, if the curved surface control point and the trimmed boundary ridge line are separately deformed, the consistency between the curved surface and the boundary ridge line is lost. Therefore, when the curved surface control point is moved directly or the shape of the boundary ridge line is deformed, it is necessary to deform while maintaining the consistency as the trim curved surface, which is a great restriction on the shape deformation. .

  When it is desired to directly deform the ridge line using direct modeling, the control point of the curved surface must be moved under a constraint condition that the ridge line after deformation is within an allowable error range.

  In general, a closed region composed of a ridge line array on a trimmed curved surface is an N-side region. It is not easy to move the curved surface control points so that the deformed ridgeline rides within an allowable error after locally deforming the boundary ridgeline of the N-side area. Therefore, it is effective to apply a new free-form surface that can maintain geometric consistency after local deformation.

  In general, the shape model of a standard specification part often includes a cutout shape as shown in FIG. FIG. 18A shows a three-dimensional model of a standard part. FIG. 18B shows a control point C10 extracted from one surface S1 of FIG. From FIG. 18B, the surface S1 is expressed as a notch shape in which the original curved surface is notched.

  Methods for applying a curved surface to an N-side area are roughly classified into a curved surface interpolation method base and an N-side filling method base (Non-Patent Document 1). The curved surface interpolation method base is a method of generating a free curved surface in which the boundary of the closed region and the boundary curve of the free curved surface coincide. The N-side Filling method base is a method of generating a quadrilateral trimmed surface that encompasses a closed region surrounded by a boundary curve.

  First, a method based on a curved surface interpolation method will be described. Non-Patent Literature 2 proposes a curved surface fitting method in which an angle tolerance ε is introduced and the angle of a normal vector at an arbitrary point on a shared boundary ridge line between adjacent patches is less than ε. Further, Non-Patent Document 3 proposes a method in which the “curved surface fitting method” of Non-Patent Document 2 is extended to handle even rational curves. Non-Patent Document 4 proposes a T-spline that relaxes restrictions on the phase of NURBS and thins out control points that are not geometrically important. Since T-spline allows T-shaped connection, control points can be greatly reduced.

  In these proposed methods of Non-Patent Documents 2 to 4, a center point and a dividing curve are generated inside the N-side area based on the subdivision method of Non-Patent Document 5, and the N-side area is divided into 4 pieces. This is a method of dividing a polygonal surface and fitting a curved surface to each quadrilateral region. For example, the N-side area R5 shown in FIG. 19 will be described. The N-side region R5 is composed of five convex ridgelines L1 to L5 that define the outer periphery thereof and one ridgeline L6 that defines a notch shape. Based on the above method (Non-Patent Documents 2 to 4), when a curved surface is applied to the N-side region R5 in FIG. 19, the internal curve L ′ having a distorted shape depending on the shape of the boundary edge L6 representing the closed region is Generated. As a result, a distorted curved surface is generated. In addition, there is a problem that it is not possible to deal with when a hole is included.

  Non-Patent Document 6 proposes a method of fitting a curved surface to an arbitrary N-side area by dividing the N-side area into an N-side patch and a quadrilateral patch that can be controlled by a star-shaped parameter f.

The method of Non-Patent Document 6 is a method in which a boundary ridge line representing a star-shaped N-side area is input and a plurality of patches are applied to the inside of the area. As shown in FIG. 20, n regular patches X _n composed of quadrilaterals are generated around a star-shaped N-side patch 201. Since the patch X _n configured by the quadrilateral holds the common area L with the adjacent patch, continuity between the quadrangular patches is maintained.

  The method of Non-Patent Document 6 enables curved surface fitting in consideration of continuity between patches by matching the boundary between patches with B-spline (hereinafter also referred to as B-spline). In addition, the size of the star-shaped N-side patch 201 generated inside the region can be controlled by the parameter f, and curved surface fitting can be applied to any N-side shape. FIG. 21 is a diagram in which the shape is controlled by changing the value of f with respect to the region represented by the same boundary ridgeline. As shown in FIG. 21B, when the value of f is reduced, the internal N-side patch 201 becomes small, and when the value of f is increased as shown in FIG. 21C, the internal N-side patch 201 becomes large. . That is, an intuitive shape operation is possible.

  Next, a trim surface-based method will be described. Patent Document 1 and Non-Patent Document 7 propose a method in which one bicubic B-spline curved surface covering the N-side area is applied to the N-side area.

  The methods of Patent Document 1 and Non-Patent Document 7 use a B-spline curved surface that covers an N-sided region and sample points based on a tangent plane, and use a curved surface fitting method (Non-patent Document 8) to create a curved surface. Fit to N-sided region. Specifically, as shown in FIG. 22A, the curved surface fitting to the N-sided region 210 surrounded by the boundary ridgelines B11 to B14 is performed by the following steps Step 1 to Step 5.

(1) About Step 1 As shown in FIG. 22A, four boundary planes 221, 222, 223, and 224 surrounding the boundary edge lines B11 to B14 are obtained. A plane is generated using the unit normal vector n calculated from the center of the boundary ridge line sequence of the N-side region 210 and the boundary ridge line sequence. Then, the boundary ridge line row is projected on the plane, and the bounding box 230 is obtained on the XY plane. With the unit normal vector n as the Z axis, four boundary planes 221, 222, 223, and 224 parallel to the Z axis are obtained from the bounding box 230.

(2) About Step 2 From the sample points on the boundary ridgelines B11 to B14 and the boundary crossing derivative at that point (Non-Patent Document 9), a straight line is generated in the direction crossing the boundary, and the straight line and the boundary planes 221, 222, The intersection points 241, 242, 243 and 244 of 223 and 224 are calculated. As shown in FIG. 22B, the sample points on the boundary edge lines B11 to B14 and the line segments having the intersection points 241, 242, 243, and 244 of the straight line and the boundary plane as both end points are the transverse line segments 251, 252, and 253. , 254.

(3) Step 3 As shown in FIG. 22 (c), the end points of a plurality of transverse lines on each boundary plane 221, 222, 223, 224 are interpolated with a B-spline curve (Non-Patent Document 8). A quadrilateral area 210A surrounding the N-side area 210 is generated. The generated quadrilateral area 210A becomes boundary curves 261, 262, 263, and 264 of B-spline curved surfaces to be applied.

(4) Step 4 As shown in FIG. 22D, a plurality of points 271, 272, 273, 274 (hereinafter referred to as point groups) are generated on the transverse line segments 251, 252, 253, 254. . The generated point group rides on the tangent plane at the sample point on the boundary ridge line.

(5) About Step 5
Using the boundary curves 261, 262, 263, and 264 of the B-spline surface generated in Step 3 and the point group generated in Step 4, the internal control points of the B-spline surface are derived by the least square method (non-patented). Reference 8).

JP 2003-330979 A

Y. Muraki, K. Konno, Y. Tokuyama: “A STUDY OF SUBDIVISION METHOD TO THREE AND FIVE SIDED FACES BASED ON REGULAR POLYGON”, Proc. Of IWAIT2009, (2009) L. Piegl, W. Tiller: “Fillingn-sided regions with NURBS patches”, The VisualComputer, Vol.2, No.15, pp.77-89, (1999) Yi-Jun. Yang, Jun-Hai. Yong, Hui. Zhang, Jean-Claude. Paul, Jia-Guang. Sun: “A rational extension of Piegl's method for filling n-sided holes”, ComputerAided Design, Vol.38, No .11, pp.1166-1178, (2006) T. W. Sederberg, D. L. Cardon, G. T. Finnigan, N. S. North, J. Zheng, and T. Lyche: “T-spline Simplification and Local Refinement”, SIGGRAPH 2004, Vol. 23, No. 3, pp. 276-283, (2004) E. Catmull, J. Clark “Recursively generated B-spline surfaces on arbitrary topologicalmeshes”, ComputerAided Design, Vol.10, No.6, pp.350-355, (1978) N. Pla-Garcia, M. Vigo-Anglada, J. Cotrina-Navau: “N-sided patches with B-spline boundaries”, Computers & Graphics, Vol.30, No.6, pp.959-970, (2006) Y. Tokuyama, K. Konno: “Filling N-sided Region with a B-spline Surface”, Information Processing Society of Japan, Vol. 43, No. 10, pp. 3209-3218, (2002) L. Piegl, W. Tiller: TheNURBS Book, Springer-Verlag, (1995) K. Konno, Y. Tokuyama, H. Chiyokura: “AG1 connectionaround complicatedcurve meshesusing C1 NURBS Boundary Gregory Patches”, Computer AidedDesign, Vol.33, No.4, pp.293-306, (2001) K. Ueda: “Meannormal vectorto a surface boundedby Beziercurves”, ComputerAided GeometricDesign, Vol.13, No.5, pp.441-451, (1996) G. Farin: Curvesand Surfaces for ComputerAided GeometricDesign A Practical Guide, Academic Press, (1996)

In the method of Non-Patent Document 6, curved surface fitting can be applied to a shape such as a star-shaped polygon. However, it is difficult to apply the method of Non-Patent Document 6 to the N-sided region 310 having a concave shape such as an L shape shown in FIG. Here, the N-side area 310 is a convex ridge line in which four ridge lines L11 to L14 swell to the outside of the N-side area, and the two ridge lines L15 and L16 are N-side shapes. A corner 400 is formed to enter the region inside. Hereinafter, this corner is referred to as a concave vertex 400. A notch portion is formed by these ridgelines L15 and L16.
Further, when the method of Non-Patent Document 6 is applied to a shape including a hole, there is a problem that an N-side patch cannot be generated inside, and as a result, a curved surface cannot be generated.

  When the method of Non-Patent Document 7 is applied to the N-side region 310 having the concave shape shown in FIG. 23, there is a problem that the generated curved surface 350 is distorted as shown in FIG. Also, Document 7 does not describe a processing method for a shape including a hole.

The following two (I, II) are considered to be the causes of the distortion of the shape of the generated curved surface.
(I) A point group (see reference numerals 271 to 274 in FIG. 22) generated from a transverse line segment (see reference numerals 251 to 254 in FIG. 22) is generated inside the region in the vicinity of the concave vertex.
(II) The method of generating the boundary ridge line of the B-spline curved surface is limited and does not correspond to the N-side area including the concave shape.

  Regarding (I) above, as shown in the circles C1 and C2 in FIG. 25A, the transverse line segments 351 and 352 extending from the sample points on the two ridgelines L15 and L16 connected to the concave vertex 400 are It extends to the inside of the N-side area 310. That is, the cross-boundary derivative is twisted. Thereby, a point cloud generated from the transverse line segment is also generated inside the N-side region 310.

  Regarding (II) above, as shown in circles C3 and C4 in FIGS. 25 (b) and 25 (c), the transverse line segments 370 and 380 generated from the concave boundary ridgelines L17 and L18 intersect with each other. May be a twisted relationship. For this reason, when a boundary curve is generated from a sequence of points obtained from the end points of the transverse line segment, the boundary curve is distorted.

  Because of the above causes (I, II), the technique of Non-Patent Document 7 can generate an undistorted B-spline curved surface for an N-sided region consisting only of convex boundary ridgelines, as shown in FIG. A distorted curved surface is generated for the N-side regions 311 to 312 including such a concave shape.

  Next, considering the shape and the posture of the space, in the methods of Patent Document 1 and Non-Patent Document 7, the four boundary planes depend on the posture of the boundary curve sequence in the three-dimensional space. The results obtained vary depending on the attitude of the three-dimensional model.

  The present invention has been created in view of the above points, and an object thereof is to provide a curved surface generation apparatus and a curved surface generation program for generating a curved surface to be applied to an N-side region including a concave shape.

In order to achieve the above object, a first configuration of the present invention is a curved surface generation device that generates a curved surface to be applied to an N-sided region including a concave shape on the surface of a three-dimensional model. A plane generation unit that sets a coordinate system and creates a boundary plane that defines the range of the curved surface to be generated, and a line segment generation that generates a transverse line segment extending from a plurality of points on the boundary ridgeline of the N-sided region to the boundary plane And the determination part that determines whether the following items (G1) and (G2) are applicable, and the matter (G1) when the determination part determines that it corresponds to the item (G1) or (G2) Or, delete the line segment belonging to the boundary edge related to (G2), set an offset curve for the boundary edge line, and recreate a new line segment with this offset curve and the boundary edge line Curve that fits the N-side area from the intersection of the part, the transverse line segment, and its boundary plane A boundary curve generator for generating a boundary curve, and a curved boundary curve from respective points on the transverse lines and surface generation unit for generating a curved surface to fit the N side-type region, comprising the.
G1: The boundary ridge line is an edge that forms a corner that enters the inside of the N-sided region.
G2: Transverse line segments extending from a point on the boundary ridge line generated based on the line segment generation unit intersect each other.
In the present specification, a shape including at least one notch portion, a hole, a concave vertex and a concave ridge line described later is generically defined as a concave shape.

  In the curved surface generation device of the present invention, it is preferable that the boundary curve generation unit generates the boundary curve based on a transverse line segment belonging to a boundary ridge line not related to the matters (G1) and (G2).

In order to achieve the above object, a second configuration of the present invention is a curved surface generation program for generating a curved surface to be applied to an N-sided region including a concave shape on the surface of a three-dimensional model. Sets a local coordinate system based on, and generates a boundary plane that defines the range of the curved surface to be generated, and generates a transverse line segment extending from a plurality of points on the boundary ridgeline of the N-side region to the boundary plane Line segment generator, Judgment unit that decides which of the following items (G1) and (G2) is applicable, and when the decision unit determines that the item (G1) or (G2) is applicable, the matter (G1 ) Or (G2), delete the line segment belonging to the boundary edge line, set an offset curve for the boundary edge line, and regenerate a new line segment with this offset curve and the boundary edge line. N-sided region from intersection of component part, transverse line segment and its boundary plane Boundary curve generator for generating a boundary curve of the curved surface fitting is characterized in that to function as the surface generation unit, which generates a curved surface fitting to the N side-type region from points on the boundary curve and the cross segments.
G1: The boundary ridge line is an edge that forms a corner that enters the inside of the N-sided region.
G2: Transverse line segments extending from a point on the boundary ridge line generated based on the line segment generation unit intersect each other.

  In the curved surface generation program of the present invention, preferably, the boundary curve generation unit generates a boundary curve based on a transverse line segment belonging to a boundary ridge line not related to the items (G1) and (G2).

  According to the present invention, in generating a curved surface of an N-sided region having a concave shape, a transverse line segment near the concave vertex is reproduced. As a result, it is possible to generate a curved surface that can be appropriately fitted and inserted into the N-side area.

  Furthermore, conventionally, the concave shape is represented by a trimmed curved surface, and the direct modeling that directly deforms the concave shape is a limited constraint that does not break the geometrical consistency between the boundary ridge line sequence and the original curved surface. However, using the direct modeling method based on the curved mesh method for trimmed curved surfaces of the present invention makes it possible to easily reconstruct trimmed curved surfaces. The degree of freedom for changing the shape of the boundary ridgeline, which has many restrictions, is expanded by the present invention, and a remarkable effect is obtained.

1 is a block diagram of a curved surface generation apparatus 1 according to an embodiment of the present invention. It is a figure for demonstrating the function for the plane production | generation part based on embodiment of this invention. It is a figure for demonstrating the function for the line segment production | generation which concerns on embodiment of this invention. It is a figure for demonstrating the function for the line segment production | generation which concerns on embodiment of this invention. It is a figure for demonstrating the function of the determination part which concerns on embodiment of this invention. It is a figure for demonstrating the function of the determination part which concerns on embodiment of this invention. It is a figure for demonstrating the function of the line segment reconstruction part which concerns on embodiment of this invention. It is a figure which shows the state which produced | generated the transverse line segment with the conventional method with respect to the N-shaped area | region which has a concave part. It is a figure which shows the state which produced | generated the transverse line segment with the method of this invention with respect to the N-shaped area | region which has a recessed part. It is a figure for demonstrating the function of the line segment reconstruction part which concerns on embodiment of this invention. It is a figure for demonstrating the function of the line segment reconstruction part which concerns on embodiment of this invention. It is a figure for demonstrating the function of the line segment reconstruction part which concerns on embodiment of this invention. It is a figure for demonstrating the function of the line segment reconstruction part which concerns on embodiment of this invention. It is a figure for demonstrating the function of the line segment reconstruction part which concerns on embodiment of this invention. It is a figure for demonstrating the function of the boundary curve production | generation part which concerns on embodiment of this invention. It is a figure for demonstrating the function of the boundary curve production | generation part which concerns on embodiment of this invention. It is a flowchart of the curved surface production | generation apparatus in embodiment of this invention. It is a figure which shows a general shape model. It is a figure which shows the N-side area | region which has a recessed part. It is a figure for demonstrating the conventional curved surface interpolation method. It is a figure for demonstrating the conventional curved surface interpolation method. It is a figure for demonstrating the conventional curved surface interpolation method. It is a figure for demonstrating the problem of the conventional curved surface interpolation method. It is a figure for demonstrating the problem of the conventional curved surface interpolation method. It is a figure for demonstrating the problem of the conventional curved surface interpolation method.

  Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings where necessary.

FIG. 1 is a block diagram of a curved surface generation apparatus 1 according to an embodiment of the present invention.
The curved surface generation apparatus 1 includes a plane generation unit 10, a line segment generation unit 20, a determination unit 30, a line segment reconstruction unit 40, a boundary curve generation unit 50, and a curved surface generation unit 60.
Hereinafter, the configuration will be described.

As shown in FIG. 2, the plane generation unit 10 creates four boundary planes 111 to 114 that define the range of the curved surface. Here, the curved surface is a B-spline curved surface applied to the N-sided region R on the surface of the three-dimensional model.
In the present embodiment, the plane generation unit 10 generates four boundary planes 111 to 114 that do not depend on the posture in the three-dimensional space where a three-dimensional model such as shape data exists. The plane generation unit 10 of the present embodiment first sets a local coordinate system (dx axis, dy axis, dz axis), and then generates four boundary planes 111 to 114.

The local coordinate system is set according to the following procedures (A1) to (A5).
(A1) The average plane 110 is generated using the unit normal vector n calculated from the center of the boundary ridgelines B1 to B6 of the N-side region R and the boundary ridgeline array (Non-patent Document 10). The average plane 110 is a plane parallel to the dx axis and the dy axis.
(A2) The boundary ridge line sequence is projected onto the average plane 110 to generate a projected boundary ridge line sequence.
(A3) Select the longest edge from the projected boundary edge line. As shown in Expression (1), the X axis (dx axis in FIG. 2) is determined using the start point p ₀ and the end point p ₁ of the selected ridge line.

(A4) As shown in the following formula (2), the cross product vector of the vectors X and n is calculated, and the Y axis (dy axis in FIG. 2) is determined. Thereby, the XY coordinate axes of the local coordinate system are set on the average plane.

(A5) A local coordinate system including the XYZ axes is set with the set XY coordinate axes and the unit average normal vector n as the Z axis (dz axis in FIG. 2).

The plane generation unit 10 of the present embodiment obtains the bounding box B ′ of the projected ridge line sequence based on the local coordinate system. In FIG. 2, the bounding box B ′ is surrounded by four lines 110a, 110b, 110c, and 110d. The four boundary planes 111 to 114 are obtained using the bounding box B ′ and the dz axis. Specifically, the boundary planes 111 to 114 are configured by sweeping lines in the dz-axis direction from the lines 110a, 110b, 110c, and 110d constituting the bounding box B ′.
The obtained four boundary planes 111 to 114 are independent of the posture of the boundary curve sequence in the three-dimensional space. That is, the method of obtaining the boundary planes 111 to 114 is affine invariant.

Next, the line segment generation unit 20 will be described.
As shown in FIG. 3, the line segment generation unit 20 generates transverse line segments 115 extending from the plurality of points on the boundary ridge lines B1 to B6 of the N-side region R to the boundary planes 111 to 114. Specifically, the line segment generation unit 20 sets a plurality of sample points SP on each of the boundary ridgelines B1 to B6 as in the related art, and a boundary crossing derivative at each sample point SP (Non-Patent Document 9). Then, a straight line is generated in a direction crossing the boundary, that is, the boundary ridge lines B1 to B6, and intersection points 111B to 114B of the straight line and the boundary planes 111 to 114 are calculated. As shown in FIG. 3, a line segment having the sample points SP on the boundary ridge line and the intersection points 111 </ b> B to 114 </ b> B of the straight line and the boundary plane is defined as a transverse line segment 115.
Further, as shown in FIG. 4, the line segment generation unit 20 generates a point group obtained by dividing each cross line segment 115 into, for example, four as sample points SP ′. This sample point SP 'is used for curved surface fitting.

Next, the determination unit 30 will be described.
The determination unit 30 determines which of the following matters G1 and G2 is applicable.
G1: border edges B1 to B6 are edges constituting corners that enter the inside of the N-side region R.
G2: The straight lines extending from the sample points SP of the boundary ridge lines B1 to B6 generated based on the line segment generation unit 20, that is, the transverse line segments 115 intersect each other.

The determination unit 30 according to the present embodiment determines the determination item G1 as follows.
For example, as shown in FIG.
As shown in FIG. 5B, v1, v2 are unit tangent vectors of the vertex P at the two ridgelines B5, B5 sharing the vertex P. The outer product vector V1 of v1 and v2 is obtained as shown in Equation (3). As shown in the equation (4), the inner product of the vector V and the average normal vector n of the boundary edge line sequence of the N-side region R is calculated, and if the inner product value is positive, a convex vertex is obtained. Let it be a vertex.


Thus, when even one of the concave vertices is present in the boundary ridgelines B1 to B6, the boundary ridgeline is determined as an irregular line. With respect to the N-side area shown in FIG. 2, the boundary ridgelines B5 and B6 are determined to be irregular lines.

Moreover, the determination part 30 of this embodiment determines the determination matter G2 as follows.
For example, all the boundary ridgelines B1 to B6 and the transverse line segment 115 are projected onto the XY plane defined by the local coordinate axes for which the boundary planes 111 to 114 are obtained, that is, the plane 110 in FIG. Next, interference points between all the transverse line segments are obtained.
Here, an N-side area R1 shown in FIG. 6A will be described as an example. The N-side area R1 includes a concave boundary ridgeline B7 that is curved so as to enter the inside. FIG. 6B shows a state in which the sample point SP is set to the concave boundary ridge line B7 and the transverse line segment 115 is generated. As indicated by an arrow α in FIG. 6B, there are at least one interference point K on the plurality of transverse line segments 115 generated from the concave boundary ridgeline B7 of the N-side region R1. become. The determination unit 30 determines that the boundary ridge line B7 to which the transverse line segment 115 having such an interference point K belongs is also an irregular line.

Next, the line segment reconstruction unit 40 regenerates a transverse line segment of the boundary ridge line determined as an irregular line by the determination unit 30 described above. Specifically, all the transverse line segments belonging to the boundary ridge line generated by the line segment generation unit 20 and determined as the irregular line are deleted, and the transverse line segment is regenerated based on the boundary transverse derivative described later. To do.
With respect to the N-side region R shown in FIG. 2, since the boundary ridgelines B5 and B6 are regarded as irregular lines, all the transverse line segments 115 belonging to these boundary ridgelines B5 and B6, more specifically β1 in FIG. All of the transverse line segments 115 to β4 are deleted, and the line segment reconstruction unit 40 regenerates the transverse line segments.

For example, as shown in FIG. 7, the outer product of the tangent plane normal vector a and the tangent vector b at each sample point on the boundary ridge line is obtained as shown in Equation (5). The obtained vector c is set in the direction of the transverse line segment. Finally, intersections between the direction vector of the transverse line segment and the four boundary planes are obtained, and a new transverse line segment is generated at the closest intersection and the sample point.

When the above operation is applied to the N-side regions R, R1, R2 having the concave shape shown in FIG. 8, the transverse line segments belonging to the irregular lines of each N-side region R, R1, R2 are shown in FIG. So that it is regenerated.

Next, reconstruction of the transverse line segment in the line segment reconstruction unit 40 of the present embodiment will be described in detail.
As shown in FIG. 10, when one of the end points of the boundary ridgelines B5 and B6 is a concave vertex 400, the Tangent Ribbon (Non-Patent Document 11) is twisted, that is, the cross-boundary derivative of Non-Patent Document 9 is from a certain position. Derived in the opposite direction. In the case of the N-sided region R in FIG. 10, as shown by reference characters β <b> 2 and β <b> 4 in FIG. 10, the boundary crossing derivative a ₀ near the concave vertex 400 is generated toward the inside of the N-sided region R. . Accordingly, in order to generate a cross-boundary derivative from the arbitrary sample point SP on the boundary ridgelines B5 and B6 including the concave vertex 400 toward the outside of the region, in this embodiment, the basic patch method (Non-patent Document 9). Is used.

FIG. 11 shows two curved surfaces S ¹ and S ² to be connected.
The cross-boundary derivative obtained from a point on the boundary edge needs to satisfy equation (6).

Where k (v) and h (v) are scalar functions of v, S ² _u is a differential vector in the u direction on the curved surface S ² , S ¹ _u is a differential vector in the u direction on the curved surface S ¹ , and S ¹ _v is a curved surface. It is a differential vector in the v direction at S ¹ .
By substituting v = 0 and v = 1 into equation (6), equation (7) is obtained.

However, k ₀ , k ₁ , h ₀ , and h ₁ are real numbers. In this method, assuming that h ₀ = 0 and h ₁ = 0, Expression (8) is obtained.

Here, in order to satisfy Expression (7), the scalar function k (v) for v is assumed to be a linear expression as shown in Expression (9).

The following equation (10) is obtained from the equations (6) and (9).

Equation (10) can be rewritten as the following equation (11) using the vector between the curved surface control points shown in FIG. However, B ³ _i (v) is a Bernstein basis function.

Since the right side of Equation (11) is a quartic equation, the order of the a _i portion is limited to the second order. Specifically, a ₀ and a ₂ are obtained from the boundary, and a ₁ is obtained by the following equation (12).

From the above expression, the boundary crossing derivative S ² _u (0, v) is expressed by the following expression (13).

In the concave vertex 400 in the N-side region R of FIG. 10, the cross-boundary derivative is generated in the reverse direction, and therefore, expressed as S ¹ _u (1, 0) = − a.
Substituting v = 0 into equation (13) yields equation (14).

From the equation obtained above, the cross-boundary derivative of the point on the boundary edge is obtained. An offset curve is generated by normalizing the cross-boundary derivative S ² _u (0, v) obtained from each point on the boundary edge and using a point group extended by the same length t. If the point at the parameter v on the boundary edge is Cv (v), the point P (v) used for generating the offset curve is expressed by the following equation (15).

  The line segment reconstruction unit 40 of the present embodiment is based on the above equation (15) for each of the boundary ridgelines determined to be irregular lines, for example, the boundary ridgelines B5 and B6 of the N-side region R in FIG. Thus, as shown in FIG. 12, auxiliary lines, that is, offset curves SL1 and SL2 are generated at positions away from the boundary ridgelines B5 and B6 in the N-side region outward direction. Specifically, the end point set regenerated based on the equation (15) is approximated by a B-spline curve to obtain an offset curve.

  Further, the line segment reconstruction unit 40 is located between the offset curves SL1 and SL2 and the corresponding boundary ridgelines B5 and B6, as shown in FIG. Draw a straight line extending to Thereby, the transverse line segment 115 'is regenerated.

  Further, the line segment reconstruction unit 40 generates, for example, a point group obtained by dividing all the regenerated transverse line segments 115 ′ into four sample points. This sample point is used for curve fitting.

  Here, when applied to an L-shaped notch shape, for example, a shape having a concave vertex, as indicated by an arrow β5 in FIG. Occurs. If the generated transverse line segments 115 'interfere with each other, the offset curve is distorted. Therefore, when the regenerated transverse line segments 115 ′ interfere with each other, the line segment reconstruction unit 40 of the present embodiment only performs the intersecting line segment 115 ′ as indicated by arrow β 6 in FIG. Is deleted. For example, in the line segment reconstruction unit 40, the method for determining whether or not the transverse line segments 115 ′ interfere with each other is projected onto the XY plane 110 and the interference point K is determined as described above. Judge by existence.

  The line segment reconstruction unit 40 of the present embodiment further performs data supplement processing. This is because, for example, when the intersecting transverse line segment is deleted as shown in FIG. As shown in FIG. 14B, the line segment reconstruction unit 50 performs a reverse search from the offset curve near the concave vertex where data shortage occurs, and regenerates the transverse line segment. For example, a point 401 having the highest curvature on the offset curve and a transverse line segment 115 ″ connecting the concave vertex 400 to the end point are added, and a point that further divides the line segment into, for example, four equal parts is generated as a sample point.

  As described above, the line segment reconstruction unit 40 of the present embodiment first extends the boundary ridge line near the concave vertex from the point on the boundary ridge line including the concave vertex by the same length based on the boundary transverse derivative. An offset curve is generated by generating a minute and then approximating the end point set of the generated line with a B-spline curve. That is, the offset curve is generated based on the tangent plane at the boundary ridge line. In this method, by considering the offset curve as Tangent Ribbon (Non-patent Document 11) and generating a transverse line segment, a point cloud can be generated outside the closed region even in the vicinity of the concave vertex.

Next, the boundary curve generation unit 50 will be described.
The boundary curve generation unit 50 generates four boundary curves of a B-spline curved surface. The boundary curve generation unit 50 does not use a transverse line segment belonging to a boundary ridge line that defines a concave shape. The boundary curve generation unit 50 generates a boundary curve using only a transverse line segment obtained from a line bulging outward (hereinafter referred to as a convex ridge line). That is, the crossing line segment belonging to the boundary ridge line determined to be an irregular line by the determination unit 30 is not used in the present embodiment. This is because the method of Non-Patent Document 7 is such that the ridge line including the concave vertex or the transverse line segment generated from the concave ridge line intersects or twists, and is obtained by the intersection of such a transverse line segment and the boundary plane. This is because there is a problem that the boundary curve is distorted because the row is wavy or looped.
Therefore, in the present embodiment, the distortion of the boundary curve is avoided by using only the transverse line segment obtained from the convex ridge line.

An N-sided region R having an L-shaped concave shape will be described as an example. FIG. 15A shows the result of obtaining the boundary lines 151 to 154 of the B-spline curved surface by obtaining the transverse line segment from the convex ridge line excluding the vicinity of the concave vertex.
Here, since the four B-spline curves 151 to 154 are independent curves, the end portions of the curves 151 to 154 coincide with each other at the corner 160 where the boundary planes intersect with each other, and are closed regions. Must be generated.

  As shown in FIG. 15 (b), the boundary curve generation unit of the present embodiment sets the end points of the boundary curves 151 to 154 to the adjacent boundary planes in order to match the end points of the boundary curves 151 to 154, respectively. Extend in the tangential direction. For example, assuming that points on the boundary plane are P1 and P2, a point P3 obtained by averaging these points is set as the end points of the boundary curves 151 to 154. Thereafter, a point sequence obtained by adding the point P3 is given to the end point on the boundary plane in the transverse line segment, and the fitting is performed again. FIG. 16 shows the result of generating a closed boundary curve. By avoiding the ridgeline connected to the concave vertex and the end point of the transverse line generated from the concave ridgeline, the problem of boundary curve self-interference is solved.

Next, the curved surface generation unit 60 will be described.
The curved surface generation unit 60 includes boundary ridgelines 151 to 154 of the B-spline curved surface generated by the boundary curve generation unit 50 and sample points and line segment reconstruction unit 40 on the transverse line segment generated by the line segment generation unit 20. Using the sample points on the transverse line regenerated by, the internal control points of the B-spline surface are derived by the least square method to generate the B-spline surface.

  The curved surface generation device 1 described above is constituted by a computer, for example. This computer implements the above-described method, that is, the generation of a B-spline curved surface, by executing a curved surface generating program as software installed in advance. Specifically, when the computer executes the curved surface generation program, the computer executes the above-described plane generation unit 10, line segment generation unit 20, determination unit 30, line segment reconstruction unit 40, and boundary curve generation unit. 50 and functions as the curved surface generation unit 60. Note that the program is downloaded via a communication line or is stored in a storage medium such as a CD-ROM, DVD-ROM, or portable storage device.

  A plurality of computers are connected to each other via a LAN, the Internet, a public network, etc., and a plane generation unit 10, a line segment generation unit 20, a determination unit 30, a line segment reconstruction unit 40, a boundary The operations of the curve generation unit 50 and the curved surface generation unit 60 may be distributed by a plurality of personal computers. A computer having a conventionally known configuration can be used, and stored in a storage device such as a storage device such as a RAM, a ROM, and a hard disk, an operation device such as a keyboard and a pointing device, and an instruction from the operation device A central processing unit (CPU) for processing data and software, a display for displaying processing results, and the like are provided. This computer may be a general-purpose device or a dedicated device.

Next, operation | movement of the curved surface production | generation apparatus 1 which concerns on embodiment of this invention is demonstrated using FIG.
First, an N-side area is designated (step S1). This specification is made by the user. For example, a computer executes a program to display a target three-dimensional model on a display, and a user operates a mouse or the like to specify an N-side area. The specified N-side area data is stored in a storage unit such as a RAM or a hard disk.

As described above, the computer starts processing of the curved surface generation program when the N-side area is designated.
First, the plane generation unit 10 sets a local coordinate system based on N-side area data arranged in the default XYZ coordinate system (step S2).

  The plane generation unit 10 creates a boundary plane (step S3). Specifically, the plane generation unit 10 projects each boundary ridge line of the N-sided region on the XY plane of the local coordinate system to create a bounding box (boundary box circumscribing the projection curve row). For example, the bounding box is a quadrilateral parallel to the X and Y axes. Then, the plane generation unit 10 generates four boundary planes by sweeping each side of the bounding box in the Z-axis direction.

  The line segment generation unit 20 generates a transverse line segment based on the boundary curve of the N-sided region (step S4). The line segment generation unit 20 first generates a plurality of sample points on each boundary curve. Next, the line segment generation unit 20 generates a straight line in a direction crossing the boundary ridge line from each sample point and the boundary crossing derivative at the sample point (Non-Patent Document 9). Furthermore, the line segment generation unit 20 calculates the intersection of this straight line and the boundary plane. Then, a line segment having both sample points on the boundary ridge line and the intersection of the straight line and the boundary plane, that is, a transverse line segment is generated. The line segment generation unit 20 generates, for example, a point group obtained by dividing the transverse line segment into four as sample points.

The determination unit 30 determines the determination items G1 and G2 (step S5).
G1: Each boundary ridge line is an edge that forms a corner that enters the inside of the square region.
G2: The straight lines extending from the sample points of the boundary ridge line generated based on the line segment generation unit 20 intersect each other.

  When the determination unit 30 determines that the boundary ridge line is an irregular line (when determined to be NG in step S5), the present embodiment performs an exception process (step S6). That is, the line segment reconstruction unit 40 recreates a transverse line segment related to the boundary ridge line determined to be an irregular line. Specifically, for the boundary ridge line of the irregular line, without using the cross-boundary derivative function of Non-Patent Document 9 described above, the N-side region outside the boundary ridge line using the above equation (15). An auxiliary line, that is, an offset curve is created at a position away from the direction. Then, a new transverse line segment that connects the offset curve and the boundary ridge line is generated. The line segment reconstruction unit 40 generates, for example, a point group obtained by dividing all the regenerated transverse line segments into four as sample points.

  The boundary curve generation unit 50 generates a boundary curve of the B-spline curved surface (Step S7). Here, when the determination unit 30 determines that some of the boundary ridge lines are irregular lines, it is created based on a non-regular line, that is, a straight boundary line or a convex boundary ridge line that bulges outward. A boundary curve is generated using only the transverse line segment.

  The curved surface generation unit 60 is regenerated by the boundary ridgeline of the B-spline curved surface generated by the boundary curve generation unit 50, the sample points on the transverse line segment generated by the line segment generation unit 20, and the line segment reconstruction unit 40. A B-spline curved surface is generated using the sample points on the traversed line segment (step S8). Specifically, the approximate curved surface is generated as a B-spline curved surface by the least square method with the boundary curve as the boundary.

According to this embodiment, for example, the following processing is performed on an N-side area having a concave vertex and a concave ridge line, that is, an N-side area having a concave shape.
(1) When any of the end points of the boundary ridge line is a concave vertex as in the shape of FIG. 8A, a transverse line segment belonging to the boundary ridge line is generated based on the offset curve.
(2) When the boundary ridge line is a concave ridge line as in the shape of FIG. 8B, or when the transverse line segments of a plurality of boundary ridge lines interfere with each other as in the shape of FIG. 8C, the boundary ridge line The crossing segment to which it belongs is generated based on the offset curve.

In the prior art, a point group based on the tangent plane in the vicinity of the concave shape is generated inside the N-sided region. Then, the point cloud is scattered in the space due to the crossing or twisting of the transverse line segment, for example, the point cloud is generated at a place that is not a real surface, and the fitting becomes ambiguous.
On the other hand, according to the curved surface generation device 1 according to the embodiment of the present invention, when generating a curved surface of an N-sided region having a concave shape, for example, a transverse line segment near the concave vertex is reproduced. As a result, it is possible to generate a curved surface that can be appropriately fitted and inserted into the N-side area.

〔Experimental result〕
This method was applied to an N-side area representing a notch shape including a concave shape.
Table 1 below shows the results of shape evaluation on nine types of curved surface shapes including notch shapes obtained from standard parts.

The average error value which is the average of the distance between the generated curved surface and the original curved surface, the maximum error value which is the maximum of the distance, the size of the bounding box and the ratio of the maximum distance are shown. In addition, the average error value that is the average of the distance between the generated curved surface and the original boundary ridgeline, the maximum error value that is the maximum of the distance, the size of the bounding box and the ratio of the maximum distance are shown. As can be seen from Table 1, the trimmed curved surface can be approximated with good accuracy.

Although detailed above, the present invention can be implemented in various forms without departing from the spirit of the invention.
The shape of the N-side area is not limited to the illustrated example.

DESCRIPTION OF SYMBOLS 1 Curved surface production | generation apparatus 10 Plane generation part 20 Line segment production | generation part 30 Judgment part 40 Line segment reconstruction part 50 Boundary curve production | generation part 60 Curved surface generation part

Claims (4)

A curved surface generating device that generates a curved surface to be applied to an N-sided region including a concave shape of a three-dimensional model surface,
A plane generation unit that sets a local coordinate system based on the N-side area and creates a boundary plane that defines a range of a curved surface to be generated;
A line segment generation unit that generates transverse line segments extending from a plurality of points on the boundary ridge line of the N-side region to the boundary plane;
A determination unit for determining which of the following items (G1) and (G2) is applicable;
If the judgment unit determines that the above item (G1) or (G2) is applicable, the crossing line segment belonging to the boundary ridge line related to the matter (G1) or (G2) is deleted, and the offset curve for the boundary ridge line is deleted. A line segment reconstruction unit that regenerates a new transverse line segment with the offset curve and the boundary edge line,
A boundary curve generation unit that generates a boundary curve of a curved surface to be applied to the N-sided region from the intersection of the transverse line segment and the boundary plane;
A curved surface generating unit that generates a curved surface to be applied to the N-side area from each point on the boundary curve and the transverse line segment;
A curved surface generation apparatus comprising:
G1: The boundary ridge line is an edge that forms a corner that enters the inside of the N-sided region.
G2: Transverse line segments extending from a point on the boundary ridge line generated based on the line segment generation unit intersect each other.   The curved surface generation device according to claim 1, wherein the boundary curve generation unit generates the boundary curve based on a transverse line segment belonging to a boundary ridge line not related to the items (G1) and (G2). . A curved surface generation program for generating a curved surface to be applied to an N-sided region including a concave shape on the surface of a three-dimensional model,
Computer
A plane generation unit that sets a local coordinate system based on the N-side area and creates a boundary plane that defines a range of a curved surface to be generated;
A line segment generation unit that generates transverse line segments extending from a plurality of points on the boundary ridge line of the N-side region to the boundary plane;
A determination unit that determines which of the following items (G1) and (G2) is applicable,
If the judgment unit determines that the above item (G1) or (G2) is applicable, the crossing line segment belonging to the boundary ridge line related to the matter (G1) or (G2) is deleted, and the offset curve for the boundary ridge line is deleted. A line segment reconstruction unit that regenerates a new transverse line segment with this offset curve and the boundary edge line,
A boundary curve generation unit that generates a boundary curve of a curved surface to be applied to the N-side area from the intersection of the transverse line segment and the boundary plane;
A curved surface generating unit that generates a curved surface to be applied to the N-side area from each point on the boundary curve and the transverse line segment;
A curved surface generation program, characterized in that it functions as:
G1: The boundary ridge line is an edge that forms a corner that enters the inside of the N-sided region.
G2: Transverse line segments extending from a point on the boundary ridge line generated based on the line segment generation unit intersect each other.   4. The curved surface generation program according to claim 3, wherein the boundary curve generation unit generates the boundary curve based on a transverse line segment belonging to a boundary ridge line not related to the items (G1) and (G2). .