JPH0836653A - Method and device for generating free curved plane - Google Patents

Abstract

PURPOSE:To generate a curved plane in smooth connection to a quadrilateral mesh with a rational B-spline curved plane with respect to each of quadrilateral meshes each comprising B-spline curves. CONSTITUTION:A mesh data input section 1 inputs a mesh comprising B-spline curves to a shape data section 7. A tangent vector data setting section 2 sets a tangent vector to some positions of a border curve. A ruled surface generating section 3 used to make smooth connection to an adjacent mesh forms ruled surface for each mesh so that tangent vectors of a curve connecting to both ends of a curve and a designated tangent vector are continuous at a tangent plane comprising differentiation vectors of a border curve. An inner curved plane generating section 5 forms a curved plane to control a shape in the inside of the mesh. A curved plane synthesis section 6 synthesizes the curved planes into one curved plane with respect to each mesh.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method and apparatus for generating a free-form surface, and more particularly to a method and apparatus for generating a free-form surface in a three-dimensional shape including a B-spline curve. For example, it is applied to a three-dimensional solid shape processing device.

[0002]

2. Description of the Related Art Two curved surfaces bounded by Bezier curves
K.Konno, T.Takamura, and H.Chiyoku can be used as a method of constructing a tangential plane-continuous transversal boundary derivative while controlling the tangent plane.
ra "A New control method for Free-FormSurfaces w
ith Tangent Continuity and its Applications "(Sci
entific Visualization of Physical Phenomena, NMPa
trikalaKis (ed.) Springer-Verlag.1991, pp.435-456)
There is. For (rational) B-spline curves, curved surfaces, etc., see Mathematical E by DFRogers, JA Adams.
lements for Computer Graphics "(second edition Mc
Graus-Hill, 1989).

As a conventional technique, a cross-boundary derivative (CBD: Cross Boundary D) at an arbitrary point on the boundary curve.
The method of interpolating a curved mesh with Gregory patch is proposed. This method is effective for subtle deformation of the curved surface shape. However, this does not allow rational curves only with polynomial curves.

Further, as a publication relating to the above-mentioned document, there is JP-A-4-344944. In this publication,
For a boundary Gregory patch that allows a rational curve to the boundary curve, the area surrounded by the rational curve is specified in order to perform the surface interpolation by specifying the trans-boundary derivative at any point on each boundary curve. On the other hand, the determination means determines the continuity at the boundary from the boundary curve and the curve connected to each boundary curve, and the connection condition determination means determines the connection condition at the boundary from the continuity obtained by the determination means, and specifies the derivative. Means for designating a cross-boundary derivative at an arbitrary point on each boundary curve, and interpolating a region with a free-form surface from the connection condition obtained by the connection condition determining means and the cross-boundary derivative specified by the derivative designating means. To do.

[0005]

As described above, in the conventional method and apparatus for generating a free-form surface, when a curved surface is conventionally generated in such a mesh-like structure, the curve forming the mesh is a Bezier curve. Some are the main, B
Generating a smooth B-spline curved surface in a region surrounded by a spline curve is rarely performed.

The present invention has been made in view of such circumstances, and for each mesh having a quadrilateral mesh-like structure composed of B-spline curves, a B-spline curved surface, more accurately, a rational It is an object of the present invention to provide a free curved surface generation method and a device for generating a curved surface that smoothly connects meshes in a mesh by using a B-spline curved surface and making the curved surface three-dimensional.

[0007]

In order to solve the above-mentioned problems, the present invention (1) specifies planes at both ends and some positions in the middle of a B-spline curve so that those planes are at both ends. And generating a curved surface shape that is a tangent plane at some positions, and (2) interpolating the inside of the region using smooth connections at each boundary of the region, or (3) Input mesh data composed of B-spline curves, set tangent vectors to some positions of the boundary curve for each curve, and for each mesh, tangent vector of the curve connected to both ends of the curve And, the specified tangent vector is constructed as a continuous surface that is continuous on the tangent plane defined by the differential vector of the boundary curve. For each mesh, a curved surface that controls the shape inside the mesh is constructed, and each mesh is constructed. For each input, combine each curved surface into one curved surface, or (4) input the mesh data composed of B-spline curve and the tangent vector data to some positions of the boundary curve. A setting unit for setting, a ruled surface generating unit for generating a ruled surface that smoothly connects to an adjacent mesh, an internal curved surface generating unit for generating an internal curved surface, a curved surface combining unit for combining curved surfaces, and each of these units. Has a shape data part for exchanging data of B, and a B-spline processing part for calculating knot insertion, sum, product, etc., and constructs a B-spline curve by constructing curved control points satisfying a smooth connection condition. The feature is that the meshes serving as boundaries are connected smoothly.

[0008]

The free-form surface generating method and apparatus of the present invention having the above-described structure (1) specify planes at both ends of the B-spline curve and at some positions in the middle of the B-spline curve. Since a curved surface shape that becomes a tangent plane at that position is generated, it is possible to smoothly connect between meshes having a B-spline curve as a boundary by configuring a control point of a curved surface that satisfies a smooth connection condition. Become. (2) Since the inside of the region is interpolated by using the smooth connection at each boundary of the region, one smooth rational B-spline curved surface can be constructed from the curved surface group that controls the shape of the boundary and the inside. become. In addition, (3) the configuration of the control points on the curved surface satisfying the smooth connection condition results in B-
It becomes possible to smoothly connect meshes having a spline curve as a boundary, and it becomes possible to configure one smooth rational B-spline curved surface from a group of curved surfaces that control the shape of the boundary and the inside.

[0009]

Embodiments will be described below with reference to the drawings. FIG. 1 is a configuration diagram for explaining an embodiment of a free-form surface generating apparatus according to the present invention, in which 1 is a mesh data input unit, 2 is a tangent vector data setting unit, and 3 is a smooth connection with an adjacent mesh. A ruled surface generation unit, 4 a B-spline processing unit, 5 an internal curved surface generation unit, 6 a curved surface synthesis unit, 7
Is the shape data part.

The mesh data input unit 1 inputs a mesh composed of B-spline curves to the shape data unit 7. The tangent vector data setting unit 2 sets tangent vectors at some positions of the boundary curve for each curve (performed for the shape data unit 7). For each mesh, the ruled surface generation unit 3 that smoothly connects the adjacent meshes has a tangent plane configured by the tangent vector of the curve connected to both ends of the curve and the specified tangent vector by the differential vector of the boundary curve. To form a continuous ruled surface (added to the shape data section 7).

The internal curved surface generating section 5 forms a curved surface for controlling the shape inside the mesh for each mesh (added to the shape data section 7). The curved surface synthesizing unit 6 synthesizes each curved surface into one curved surface using the equation (1) for each mesh (creates and adds new data using the added data). The B-spline processing unit 4 calculates knots, sums, products, and the like. In the following description, notations such as u [*, i] correspond to those described in Table 1.

[0012]

[Table 1]

A boundary curve C of a quadrilateral mesh
(U) is a B-spline curve of d order as follows.

[0014]

[Equation 1]

The knot sequence C is C = [S ₀ , ..., S _{K + d + 1} ]
Is. At this time, as shown in FIG. 2, the tangent vectors of the curved surfaces on both sides of the curve at K + 1 parameter values u [*, i] are V [l, i], V [r, i] (i = 0, … 、
k). However, u [*,
0] = S ₀ u [*, k] = S _{K + d + 1} .

If all k + 1 parameter values u
In [*, i], the tangent vector U (u [*,
If i]) and V [l, i], V [r, i] are on the same plane, it is desirable to construct a curved surface having a specified tangent vector and being G ¹ continuous. First, the derivative U (u) of the boundary curve C (u) is

[0017]

[Equation 2]

It is a vector function of d-1 order as follows.
Then, a common tangent vector function V (u) that defines the tangent plane
Is configured. U (u [l, i]), V [l, i], V
The plane defined by [r, i] has V [l, i] + V
[R, i] is also on board, so U (u [l, i]) and V
A tangent plane is defined by [l, i] + V [r, i] (FIG. 3). After that, V [l, i] + V [r, i]
A vector function V (u) for interpolating is defined, and the plane defined by U (u) and V (u) is a tangent plane on the curve.
The function V (u) is

[0019]

(Equation 3)

Is a function such as
-Interpolate between them with the spline function.

[0021]

[Equation 4]

At this time, the knock vector T is T = [t ₀ = t ₀ = t ₁ <t ₂ <t ₃ <... t _k-1 <t _k = t _{k + 1}
= T _{k + 1} ], and t ₀ = u [*, 0] = t ₁ <u [*, 1] <t ₂ <u
[*, 2] <... u [*, k-2] <t _k-1 <u [*, k
-1] <t _k = u [*, k] = t _{k + 1} . For example,

[0023]

(Equation 5)

However, if there is a condition s _j in the knot sequence C of the boundary curve, it is possible to reduce the number of control vectors of the final tangent vector function by using that value. . As a result, the following linear equation is derived as in the case of function interpolation by B-spline.

[0025]

(Equation 6)

Since this is a tridiagonal matrix, it can be easily solved. Here, α _i β _i γ _i is the following value.

[0027]

(Equation 7)

Now that the derivative and the common vector function have been obtained along the boundary curve, it is the stage of calculating the tangent vector functions of the curved surfaces on both sides of the boundary curve.

[0029]

(Equation 8)

Since it must satisfy the condition of G ¹ continuity,

[0031]

[Equation 9]

K + 1 parameters u of these functions
The value in [*, i] is based on the following tangent plane conditions for each parameter.

[0033]

[Equation 10]

[0034]

[Equation 11]

It is Again, describing the problem,

[0036]

(Equation 12)

When W (u) = f (u) U (u) + g (u) V (u), W (u), f (u), g (u) are calculated. Is.

[0038]

(Equation 13)

It has already been determined that the condition is satisfied. First, the functions f (u) and g (u) can be C ¹ continuous functions in the following form.

[0040]

[Equation 14]

The B-spline coefficient f _i is the same as V (u),

[0042]

(Equation 15)

From, it can be obtained by solving the simultaneous equations. The same applies to the function g (u). The rest is W
In the calculation of (u), W (u) = f (u) U (u) + g (u) V (u) where f (u) and g (u) are quadratic and U (u) Is d-1
Next, V (u) is also a quadratic B-spline, so d +
First-order B-spline vector function

[0044]

[Equation 16]

It becomes This W (u) is the boundary curve C
Since it is a linear combination of the derivative U (u) of (u) and the tangent plane vector V (u), it lies on the plane defined by U (u) and V (u).

[0046]

[Equation 17]

Here, f (u), g (u), W (u),
Since V (u) is eventually in the B-spline format, f (u)
The product of B-splines such as (u), W (u), g (u), V (u) is described in the following article by Morken k. Morken,
`` Some Identities for Products and Degree Raising
of Splines ”(Constructive Approximation 7, 1991,
pp.195-208), it can be converted into one B-spline format.

Also, f (u) U (u) + g (u) V
The sum of two B-spline formats as shown in (u) becomes one B-spline format by aligning knot vectors and calculating the sum of the coefficients of each spline. After all, on both sides of the boundary curve, two B-
This means that the tangent vector function expressed in spline form has been obtained.

4 (a) to 4 (d) construct a tangential plane continuous derivative having specified tangent vectors on both sides of the cubic B-spline curve, and express the derivative with a ruled surface. It is an example. It can be seen that the tangent vector specified by the contour parameter lines is satisfied and the contour lines are connected smoothly.

It is assumed that a tangent vector as shown in FIG. 5 is obtained as a ruled surface at each boundary of the quadrilateral region surrounded by the B-spline curve by using the method described above (in FIG. 5, W [l, (u, v)] and W [r, (u,
v)] is the same, and a ruled surface showing a tangent vector is shown outside the region). By using these tangent vectors and boundary curves, the area is interpolated by a curved surface expression expressed by the following expression.

[0051]

(Equation 18)

In FIG. 5, the knot vector is T = [t ⁰ ,
t ⁰ , t ⁰ , t ¹ , t ¹ , t ² , t ² , t ² ], but U =
[U ⁰ , U ⁰ , U ⁰ , U ¹ , U ¹ , U ² , U ² , U ² ], V = [V ⁰ ,
V ⁰ , V ⁰ , V ¹ , V ¹ , V ² , V ² , V ² ] can be independently selected.

In the equation (1), the four curved surfaces U ₀ (u, v), U ₁ (u, v), V ₀ (u, v) and V ₁ (u, v) as shown in FIG. Besides, a fifth curved surface called W (u, v) is required,
For example, as shown in FIG. 7, it is possible to use the ruled surface of the boundary curve on one of the opposite sides (the shape of the curved surface for interpolating the region can be controlled by selecting this curved surface expression). As a result, the area as shown in FIG. 5 is interpolated by the curved surface as shown in FIG. The curved surface equation of equation (1) is

[0054]

[Formula 19]

Putting it aside,

[0056]

(Equation 20)

It can be expressed as follows. Product W _i (u, v) S
_{Both i} (u, v) are in the B-spline format, and can be converted into the B-spline format (according to Morken's paper), which results in the rational B-spline format. Since these have a common denominator, if the knot vectors are aligned, the sum can also be in the B-spline form. The result of this processing is shown in FIG. 9, which is the interpolated curved surface and control net in the central portion of FIG.

When the curved surface of the equation (1) is mathematically converted, the weights of the control points at the four corners of the generated rational B-spline curved surface become 0, so that the equation (1) is calculated using a minute amount ε. The control net of FIG. 9 is calculated by modifying the equation (2).

[0059]

[Equation 21]

In the case of a non-quadrilateral mesh, a plurality of four
The method of the present invention can be used by dividing the mesh into a quadrangle.

FIG. 10 is a flow chart for explaining the free-form surface generating method according to the present invention. Hereinafter, each step (S) will be described in order. First, a mesh composed of B-spline curves is input to the shape data section 7 (S1), and for each curve, setting of tangent vectors to some positions of the boundary curve is set in the shape data section. It does to (S2).

Next, for each mesh, a ruled surface is constructed so that the tangent vector of the curve connecting both ends of the curve and the specified tangent vector are continuous on the tangent plane defined by the differential vector of the boundary curve. Yes (S3). Next, a curved surface for controlling the shape inside the mesh is formed for each mesh (S4), and each curved surface is combined into one curved surface using the equation (1) for each mesh (S5).

[0063]

As is apparent from the above description, the present invention has the following effects. (1) Effect corresponding to claim 1: By designating planes at both ends of the B-spline curve and at some positions in the middle, those planes become tangent planes at both ends and some positions. Since the curved surface shape is generated, it is possible to smoothly connect the meshes having the B-spline curve as a boundary by configuring the control points of the curved surface that satisfy the smooth connection condition. (2) Effect corresponding to claim 2: Since the inside of the region is interpolated by using the smooth connection at each boundary of the region, one smooth rational B-from the curved surface group that controls the shape of the boundary and the inside. It becomes possible to construct a spline curved surface. (3) Effects corresponding to claims 3 and 4: The configuration of the control points of the curved surface satisfying the smooth connection condition makes it possible to smoothly connect between the meshes having the B-spline curve as a boundary, and also at the boundary. And a smooth rational B-spline curved surface can be constructed from the curved surface group controlling the internal shape.

[Brief description of drawings]

FIG. 1 is a configuration diagram for explaining an embodiment of a free-form surface generating apparatus according to the present invention.

FIG. 2 is a diagram for explaining designation of a tangent vector in the present invention.

FIG. 3 is a diagram for explaining determination of a tangent plane in the present invention.

FIG. 4 is a diagram for explaining generation of a ruled surface in the present invention.

FIG. 5 is a diagram showing how a tangent vector in the present invention is obtained as a ruled surface.

FIG. 6 is a diagram showing a B-spline basis function in the present invention.

FIG. 7 is a diagram showing how to use a ruled surface of a boundary curve of one opposite side in the present invention.

FIG. 8 is a diagram showing an interpolated curved surface in the present invention.

9 is a diagram showing the interpolation curved surface of FIG. 8 in a B-spline format.

FIG. 10 is a flowchart for explaining a free-form surface generation method according to the present invention.

[Explanation of symbols]

DESCRIPTION OF SYMBOLS 1 ... Mesh data input part, 2 ... Tangent vector data setting part, 3 ... Ruled surface generation part which connects smoothly with adjacent meshes, 4 ... B-spline processing part, 5 ... Internal curved surface generation part,
6 ... Curved surface synthesis section, 7 ... Shape data section.

Claims (4)

[Claims] 1. By designating planes at both ends of a B-spline curve and at some positions in the middle of the B-spline curve, it is possible to generate a curved surface shape such that the planes become tangent planes at both ends and some positions. Characteristic free-form surface generation method. 2. The free-form surface generating method according to claim 1, wherein the inside of the region is interpolated by using a smooth connection at each boundary of the region. 3. Input mesh data composed of B-spline curves, set tangent vectors to some positions of the boundary curve for each curve, and connect each mesh to both ends of the curve. The curved surface that controls the shape inside the mesh is constructed for each mesh by constructing a ruled surface so that the tangent vector of the curved line and the specified tangent vector are continuous on the tangent plane defined by the differential vector of the boundary curve. A method of generating a free-form surface, which comprises structuring and combining each curved surface into one curved surface. 4. An input unit for inputting mesh data composed of B-spline curves, a setting unit for setting tangent vector data to some positions of a boundary curve, and a line weave that smoothly connects to adjacent meshes. A ruled surface generating unit for generating a surface, an internal curved surface generating unit for generating an internal curved surface, a curved surface synthesizing unit for synthesizing curved surfaces, a shape data unit for exchanging data with these units, a knot insertion, a sum , A B-spline processing unit for calculating a product and the like, and smoothly connecting meshes having a B-spline curve as a boundary by configuring a control point of a curved surface satisfying a smooth connection condition. Characterized free-form surface generator.