JPH0380373A - Sculptured surface generating system - Google Patents

Abstract

PURPOSE:To enable a sculptured surface to be handled in stereographic shape including a rational curve by deciding continuity at a boundary, and generating the sculptured surface by generating the internal control point of a plane from a found connection condition. CONSTITUTION:The sculptured surface can be generated with a deciding part 1 which decides the continuity of the plane whose boundary is enclosed with the rational curve at the boundary from a boundary curve and a curve connected to each boundary curve, and a condition setting part 2 which finds the connection condition at the boundary from the continuity found at the deciding part 1, and a generation part 3 which generates the internal control point of the plane from the connection condition found at the condition setting part 2. In the sculptured surface generating system, a double tertiary rational boundary Gregory patch is used as the expressive quation of a curved surface. Thereby, it is possible to generate the sculptured surface on the plane having the boundary curve including three or more rational curves as keeping the continuity.

Description

DETAILED DESCRIPTION OF THE INVENTION The present invention relates to a three-dimensional three-dimensional shape processing device, and more particularly to a free-form surface generation method for a three-dimensional shape including rational curves.

When expressing a three-dimensional shape on a computer, an important issue is how to model a complex free-form surface.One method for modeling a zero-curved surface is to first draw the contour of the curved surface. There is a method of creating a curve model to represent the object and interpolating the surface from that curve model. For example, enter a cross-section line that extracts the features of a free-form surface.

There is a cross-section line input method in which the system generates a free-form surface between the cross-section lines, and a polygon input method in which the system generates a free-form surface from the polyhedron after inputting a shape that approximates the free-form surface with a polyhedron. With such a method, the designer only needs to be aware of the curve from which the features of the final shape are extracted, which is less burdensome than creating the shape of the free-form surface itself.

When a designer freely creates a curved model, the surface surrounded by the mesh may become a non-quadrilateral surface such as a triangular surface or a pentagonal surface. In addition, the created curve model may accurately represent a quadratic surface shape such as a cylindrical surface, a conical surface, or a spherical surface. In order to accurately express this quadratic surface shape, a rational curve must be used as the boundary curve.

In this way, in the process of creating a shape intended by a designer, a curve model with irregular boundary curves appears, and rational curves may be used as the curve model. The method of interpolating an irregular curve model with a free-form surface so that it becomes 01 continuous (G1 indicates the degree of smoothness and becomes 01 by reparameterization) is based on the fact that the curve model is a polynomial. This was possible only if it was expressed.

However, with this conventional method, it has not been possible to interpolate a curve model including a rational curve so that it becomes 01 continuous on a free-form surface.

Prior art documents related to the present invention include the following.

■L Piegl and Wayne Tiler
, “Curve and 5 surface construction
uctions using rational B-
splines”, computer-aided
design, Vol. 19. No, 9. May,
1987t pp201-212゜■ Written by Hiroaki Chiyokura
"Solid Modeling" (Published by Kogyo Kenkyukai, 1986-0-4-3)
0).

With such conventional techniques, a free-form surface can be generated when the boundary curve is not a rational curve.

When the area surrounded by the curved mesh shown in Figure 6 (a) and (b) is non-quadrilateral, or when the nodes of the curve are not in contact with four curves, it is called an irregular curve model. .

In particular, as shown in Fig. 7, one surface FL has two surfaces F, , F.
, is connected, node vL is connected to 7 nodes (T-no
It is called de). In order to generate a free-form surface that is G1 continuous using B ezier patches, the curved mesh must be regular, so the area surrounded by the curved mesh shown in Figures 6 and 7 is free-form. Unable to generate curved surface. To solve this problem, G r
A method has been proposed for generating a free-form surface that guarantees G1 continuity for a curve model expressed by an arbitrary Bezier curve by a curve mesh interpolation method using an egory patch. However, when creating a curve model that includes a mixture of quadratic surfaces and free-form surfaces, such as cylindrical surfaces, conical surfaces, and pentospheres, rational curves are used as part of the curve mesh. For this rational boundary mesh, the conventional Grego
If we try to derive a 01 continuous conditional expression from a rationalized expression of the ry batch, the conditional expression becomes complicated and cannot be solved easily. Similarly, for a regular rational boundary mesh, G1 derived from the rationalized formula of the Bazier patch
Since the continuity conditional expression is also complicated, it cannot be solved easily.

Figure 8 shows a quadratic surface and a Gregorian curved surface in a part of the curved mesh.
FIG. 3 is a diagram showing a plurality of types of patches including a mixture of ry patches. In this way, the curved surface formula that has existed until now,
For a rational boundary mesh, it was not possible to interpolate the surface to make it G1 continuous.

The present invention was made in view of the above-mentioned circumstances, and
The purpose of this invention is to provide a free-form surface generation method that can generate a free-form surface on a surface having a boundary curve including three or more rational curves while maintaining continuity.

Keyaki - In order to achieve the above object, the present invention includes a determination unit that determines continuity at the boundary from boundary curves and curves connected to each boundary curve for a surface surrounded by rational curves. , a free-form surface is generated by a condition setting unit that determines a connection condition at the boundary from the continuity determined by the determination unit, and a generation unit that generates internal control points of the surface from the connection condition determined by the condition setting unit. It is characterized by

In the free-form surface generation method of the present invention, a bicubic rational boundary Gragory patch is used as a surface expression. Before describing this surface generation method, the characteristics of the rational boundary Gregory patch will be explained. Here, the rational boundary G regory
The patches are assumed to be bicubic unless otherwise specified. Rational boundary G regory patch.

As shown in FIG. 9, it is expressed by 20 control points. Each control point defining the boundary curve has a different weight in each direction of the patch parameters u and v. Moreover, since the differential vector on the rational boundary curve can be easily defined, the expression for the connection between the patches sandwiching the rational curve can be easily derived. The control points of this rational boundary G regory patch are P tyJ+k (1= O+...t
3 + J = Or..., 3. =o, 1), the expression of the curved surface is as follows.

however, shall be.

Each control point of the patch and the weight at each control point.

It is expressed as below.

When i=0, 1; j=o, 1 i=:2,3; when j=o,1 i=o, 1; when j=2.3 i=2,3; when j=2.3 However, 0<u, vko1 This patch formula has the following characteristics.

■When the weight is 0 or more, it has convex closure like the Gregory patch. This is a property that any point on a curved surface exists within a region surrounded by control points of one curved surface. Due to this property, a rough check for interference between curved surfaces can be performed at high speed.

■A special case of the bicubic rational boundary Gregory patch is the bicubic rational Bezier patch. Control points PIIO and Pt+ of rational boundary G regory batch
t+p ito and PllLy Ptxo and P1ml+
When Pzxo and Pzzi are each equal and the weights at each of these control points are equal, the rational boundary G regor
The y patch will be equal to the rational B ezier patch.

■Differential vector functions that cross the boundary curve can be defined independently. Now, let us express the differential vector function in the direction of parameter U as follows.

j'' = S, (u, v) U Therefore, each differential vector function across the boundary curve is defined as follows.

This property enables local curved surface generation.

Here, we define two rational boundary G regory patches as G
A method of connecting them in one continuous manner will be described. 1st
Figure 0 represents two connected rational boundary Gregory patches S (u, v) and S' (u, v). Also, the control point of each patch is P tJ1+t P'
If tJh, then the vector a(,
btlct is defined as follows.

aL= Pitt PzLt (i=o,
..., 3) bL=P', L□-P'. 4, (
1=ol..., 3)c L=P'o, t+x,,
P'oit (i=oy..., 2) Also, in order for two patches to be connected so that they are 01 continuous, the three differential vectors at any point on the boundary curve must be on the same plane. It must be in In other words, the following formula must be satisfied.

S'? (0,v) =k(v)SJI,v)+h(v)
S, (1, v) (5) Also, the differential vector S, (1, v) on the boundary curve is expressed by the following equation.

(6) However, the functions Ti and T are expressed as follows.

It is also assumed that the functions h (v) and k (v) are scalar functions.

First, in order for two patches to be continuous on a tangent plane, the vertex V is required. , vl must be on the same plane. This can be expressed as follows.

))o=kOao+hllCG b3=to, a3+h, c ma However, k, ni, ho, and hl are scalars.

Let us now consider equation (6). Since equation (6) is complicated, it is difficult to connect the patches so that they are G1 continuous if left as is. Therefore, by adding the following to the connection conditions at the boundary curve, it becomes possible to continuously connect two surfaces that sandwich a rational curve.

■Flow vector alllbO around vertex v0 and vertex v
When the flow vectors a39b3 around 1 are on the same straight line, the equation of connection around the vertex is (5
), S', (0, v) = k(v) S, (1, v). Therefore, the conditions for two patches to be continuous on a tangent plane are that S is at most a quadratic expression and the vector between control points satisfies the following expression.

b x ” (k o W a t z a 1
+ M w 36, an) w3□,
3 bz= (koWazo, , + mW,,,
a,) w3. ．． 3 ■On the ridgeline where patches S and S' are in contact, when the weight at each control point satisfies the following relationship, equations (1), (2), and (6) become (7), respectively. ) formula, (8)
The equation can be transformed as shown in equation (9). In other words, w2J,
:= w3J□ (j=0...3)WoJ, = W
, J, (j = O...3), then the differential vector function on the boundary curve is expressed as follows.

This is a condition where the adjacent edges become non-rational curves. Substituting equations (7), (8), and (9) into equation (5),
The conditions for two consecutive patches to be 01 are as follows.

b, -'(k41W3txal+MW3ataa"2
“!”Wi,zCo”!iWioxce)woo,
33 bz=±r'N5sxa3"(kzW32ta2"?W
izzCi”?WitxCz)wo, 3 Using the above procedure, the internal control points of the curved surface can be determined so that the two patches are 01 consecutive.

Furthermore, since the differential vector function SJI,v) is a quadratic function at most, a quadratic Bezier function is first defined using three vectors aoyQta. From this function, the flow vector a□lag is determined using the following equation.

wa+2 a, -3W3x□ 2+wa a" 3 w3.

So far, we have described a method for generating G2 consecutive rational boundary G regory patches. Here, a process of generating patches on a surface as shown in FIG. 11 will be described.

(2) Definition of the flow vector function The plane F is surrounded by five rational boundary curves of Et (i=o, . . . , 5). On this boundary curve, a quadratic flow vector function (cross-boundary
tive). In this case, the vertex v1 of the ridge lines E, .

Therefore, one flow vector function g , (t) is defined around the curve tangent to the T node. In other words, T
The flow vector function g, (t) at the edge line that is in contact with the vertex V, which is a node, is expressed by the flow vector function g, (t) on the edge line E, E3. E,,E,
,E., ,El. , E1□. If, face F□
When the plane F2 and the plane F2 are connected to G1 continuously, the flow vector function gi(t) is
1. Defined from

Also, when one surface F1 and one surface F2 are connected in G1 continuity, the flow vector function g 1 (t) when generating a free-form surface on the surface F1 and when generating a free-form surface on the surface F2
are the same.

■The flow vector function set on the boundary curve of the decision surface F□ of the internal control point is
Since it is at most quadratic, the cubic vector aojyatl+
azJF = Q, ---, 3) is defined by the B ezier function. By applying these three vectors to the above-mentioned connection equation, the flow vector that determines the internal control point is determined. Here, if the surface F1 can be represented by a single batch, the control points found here define the batch itself, and the process ends.

(4) Division of surfaces When one surface is non-quadrilateral as shown in FIGS. 12(a) and (b), the surface is divided into several quadrilateral melting surfaces. After that, a rational boundary Gregory patch is generated for each divided surface. In that case, the boundary curves are rational curves, but if the internal boundary curves dividing one surface are non-rational curves, 01 continuous connections can be made around each internal boundary curve.

Hereinafter, the present invention will be explained based on examples.

FIG. 1 is a block diagram for explaining an embodiment of a free-form surface generation method according to the present invention. In the figure, 1 is a determination section, 2 is a condition setting section, and 3 is a generation section.

The determination unit 1 determines continuity at the boundary of a surface surrounded by rational curves from boundary curves and curves connected to each boundary curve.

The condition setting unit 2 determines the boundary connection condition from the continuity determined by the determination unit 1.

The generation unit 3 generates internal control points of the plane from the connection conditions determined by the condition setting unit 2.

FIG. 2 is a diagram showing a flowchart of the free-form surface generation method according to the present invention. Hereinafter, each 5 step will be explained in order.

m; Store the control points of all the boundary curves that make up the surface in the storage device. At the same time, among the curves connected to the end points of each boundary curve, the control points of the curves constituting the adjacent surface are stored in the storage device. Control points on the boundary curve are shown in FIG.

Master: 5tep Using the control points of the curve stored in step 1, determine the continuity with the adjacent surface across the boundary curve, and store the connection conditions in the storage device.

Step 3 1 3 2; 5 From the connection conditions obtained in top 2, a flow vector directed toward the control point on the boundary curve and the center of gravity of the surface is determined and stored in a storage device. The flow vector in the direction of the center of gravity of the surface is shown in FIG. The internal control points of the surface are calculated from the control points on the curve and the flow vector. Here, if the surface is a quadrangular surface, the process ends.

Oyara soil; 5tep 3-1 or 5tep 3-2
Interpolate the non-quadrilateral surface from the control points on the boundary curve obtained in and the flow vector toward the center of gravity of the surface. The interpolation results are shown in FIG.

FIG. 13 shows an example of a curve model including T nodes. A 0-face F construction has two T nodes v1 and v2. 14th
In the figure, planes Fi are connected 01 consecutively and cross-sectional lines are displayed. In FIG. 14, for example, surface A (surface F construction) formed by btGtdte is formed by surface B formed by gt he l, or surface C1a, b, f formed by a, f, e. Face and 0
1 continuous connection. Moreover, FIG. 15 is an example of an irregular curve model including a rational curve. Figure 16 shows
The surface F work shown in FIG. 15 is connected 01 consecutively and the cross-sectional line is displayed.

- As is clear from the above description, according to the present invention, a free-form surface can be generated on a surface having a rational boundary curve while maintaining continuity. In other words, as shown in Figure 8, a part of the surface where the curved mesh appears is a quadratic surface or
Even if there is an ory batch, interpolation can be performed continuously in G1.

This makes it possible to handle free-form surfaces in three-dimensional shapes including rational curves.

[Brief explanation of drawings]

FIG. 1 is a block diagram for explaining an embodiment of the free-form surface generation method according to the present invention, FIG. 2 is a flow chart thereof, FIG. 3 is a diagram showing control points of boundary curves, and FIG. The figures show the flow vector in the direction of the center of gravity of the surface, Figure 5 shows the interpolation results, Figures 6 (a) and (b) show the irregular curve model, and Figure 7 shows the interpolation results. The figure shows a curve model including the T node.
8 shows a plurality of types of patches, and FIG. 9 shows control of rational boundary Gregory patches. FIG. 10 is a diagram showing the connection of the rational boundary G regory pad. FIG. 11 is a diagram showing the rational boundary curve and the one-minute vector function. FIG. 12 (a) and (b) are the curved surfaces. Figure 13 is a diagram showing a 1-curve model including the T node, Figure 4 is Figure t3 (a diagram showing a cross-sectional diagram,
Figure 15 includes rational curves. FIG. 16 is a diagram showing an irregular curve model, and FIG. 16 is a diagram showing a cross-sectional line diagram of FIG. 115. 1... Judgment section, 2... Condition setting section, 3... Generation section.

Claims (1)

[Claims] 1. For a surface whose boundary is surrounded by several curves including rational curves, there is a determination unit that determines the continuity at the boundary from the boundary curves and the curves connected to each boundary curve, and the determination unit determines the continuity at the boundary. Free-form surface generation characterized in that a free-form surface is generated by a condition setting unit that determines a connection condition at a boundary from continuity, and a generation unit that generates internal control points of a surface from the connection condition determined by the condition setting unit. method.