JPH0696163A - Blended curved surface generator - Google Patents

Abstract

PURPOSE:To generate a blended curved surface on which the global characteristic of the shape of an initial curved surface is reflected when the initial curved surface connecting two curved surfaces successively is designated and to give certain physical interpretation on deformation from the initial curved surface to the blended curved surface. CONSTITUTION:The generator is provided with a deformation group generating means 12 which generates the deformation group of the initial curved surface when the blended curved surface is generated by deforming the initial curved surface in the normal direction, a Laplacean calculation means 13 which calculates the Laplacean of the initial curved surface, and a minimum energy deformation retrieval means 14 which retrieves the deformation to minimize deformation energy according to the Laplacean out of the deformation group generated by the deformation group generating means 12 based on a result calculated by the Laplacean calculation means 13. The deformation in which the one to minimize the deformation energy E is applied to the initial curved surface is set as the desired blended curved surface.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a blended curved surface generator which is used when a curved surface is designed by using a computer and which generates a blended curved surface for smoothly connecting two given curved surfaces.

[0002]

2. Description of the Related Art When designing a curved surface shape using a computer, a curved surface having a complicated shape is constructed by connecting several simpler curved surfaces (element curved surfaces). When designing curved surfaces in this way, element curved surfaces generally do not intersect smoothly with each other, and therefore blend curved surfaces are used when smooth connection is required. The blended curved surface is a curved surface that smoothly connects two given curved surfaces.

As a method of generating a blended curved surface, for example, CLBajaj: "Surface fitting using implicit alge"
braic surface patches, Curve and Surface Modelin
g ", SIAMPublications (1992), MIGBloor and MJWi
lson: "Generating blend surfaces using partial diff
erential equations ", Computer-Aided Design, 21 (198
9), 165-171, DJFilip: "Blending parametric surfac
es ", ACM Transactions on Graphics, 8 (1989), 164-173
As shown in, etc., consider the initial curved surface that continuously connects the two curved surfaces A and B to be connected, and consider the boundary conditions (boundary curve, normal vector or cross on the boundary) of the curved surface to be connected. (Direction of tangent vector) and parameters that roughly control the shape of the blend surface to be generated (control points, that is, the point where the blend surface passes near it, the size of the cross tangent vector on the boundary, etc.) are specified. , Target of connection 2
There is a method of generating a blended curved surface by deforming the initial curved surface X so that the curved surfaces A and B are smoothly connected.

[0004]

When the initial curved surface X is deformed to generate the blended curved surface, the conventional method cannot reflect the global geometric features of the initial curved surface X on the blended curved surface, and There is a problem that the user does not have a natural impression because the transformation from the curved surface to the blended curved surface has no physical interpretation.

An object of the present invention is, when an initial curved surface that continuously connects two curved surfaces is designated, can generate a blended curved surface reflecting the global characteristics of the shape of this initial curved surface, and from the initial curved surface. It is an object of the present invention to provide a blended curved surface generator that involves a physical interpretation of the transformation to the blended curved surface.

[0006]

In the blended curved surface generating apparatus of the present invention, an initial curved surface for continuously connecting the given two curved surfaces is designated for the given two curved surfaces. In a blended curved surface generator that generates a blended curved surface that smoothly connects between the given two curved surfaces by transforming the initial curved surface in the normal direction to generate a blended curved surface. Deformation group generation means for generating the deformation group of the initial curved surface, Laplacian calculation means for calculating the Laplacian of the initial curved surface, and deformation generated by the deformation group generation means based on the result calculated by the Laplacian calculation means. A minimum energy deformation search means for searching a deformation that minimizes the deformation energy associated with the Laplacian in the family; And a blending surface generation means for generating a blended curved by applying the deformation which is searched by the Energy deformation searching means to the initial surface.

[0007]

When the two curved surfaces A and B which are desired to be smoothly connected and the initial curved surface X which continuously connects these two curved surfaces A and B are designated, the present invention first uses the data of these curved surfaces to calculate the two curved surfaces A and B. A deformation group of the initial curved surface X that becomes a blended curved surface between B is obtained. When a blended curved surface is obtained by deforming the initial curved surface X in the direction of its normal, the manner of deformation is generally not uniquely determined. Here, a group of deformations of the initial curved surface and a group of deformations that give a blended curved surface is referred to as a deformation group. In the following description, unless otherwise specified, “deformation” does not represent a specific deforming operation but represents a deformation as a mathematical operator.

The Laplacian Δ of the initial curved surface X is a differential operator determined from the geometric shape of the initial curved surface X. Then, the deformation energy E associated with this Laplacian
(φ) is defined by the following equation.

[0009]

[Equation 1] Where φ is the deformation of the initial curved surface X in the direction of its outer normal, d
A is an area element of the initial curved surface X. The deformation energy E (φ) associated with the Laplacian Δ of the initial curved surface X has the following characteristics. The deformation energy E (φ) roughly represents the overall difference between the average curvature of the initial curved surface X and the average curvature of the curved surface obtained by subjecting the initial curved surface X to the deformation φ. When the initial curved surface X is regarded as a curved surface in the equilibrium state of a homogeneous elastic film having a fixed boundary, the deformation energy E (φ) roughly represents the magnitude of the restoring force of the deformed film.

In the present invention, a deformation that minimizes the above-mentioned deformation energy E (φ) is selected from the deformation group of the initial curved surface X, and this deformation is applied to the initial curved surface X to generate a blended curved surface. The curved surface retains the global geometric characteristics of the initial curved surface, and the concrete deformation from the initial curved surface X to the blended curved surface has a physical interpretation, so that it is possible to easily grasp the image of this specific deformation. become.

[0011]

Embodiments of the present invention will now be described with reference to the drawings. In this embodiment, a bicubic NURBS (Non-Uniform Rational B-Spline) surface, which is one kind of spline surface, is used as the given two surfaces A and B and the initial surface X. It is applicable to any parametric surface for which partial derivatives up to the third order have been calculated. FIG. 1 is a block diagram showing the configuration of a blended curved surface generator according to an embodiment of the present invention, and FIG. 2 is a diagram for explaining a modified example from an initial curved surface to a blended curved surface.

The blend curved surface generator 10 of this embodiment is
When two curved surfaces A and B that are desired to be smoothly connected to each other and an initial curved surface X that continuously connects the two curved surfaces A and B are input by the curved surface modeler 21, the initial curved surface X is set to the outer normal direction thereof. This is for generating a blended curved surface between the two curved surfaces A and B by outputting it to the curved surface display device 22. The blended curved surface generator 10 includes a curved surface modeler 21.
Parametric expression generation means 11 for obtaining the parametric expression of each curved surface A, B, X and the parametric expression of the unit outside vector field of these curved surfaces A, B, X from the data input from, and the curved surfaces A, B according to the obtained parametric expression. A deformation group generation means 12 for generating a deformation group of the initial curved surface X into a blended curved surface between them, a Laplacian calculation means 13 for calculating a Laplacian Δ and an area element dA of the initial curved surface X according to the obtained parametric expression, and a deformation group generation means 12. A minimum energy deformation search means 1 for finding a deformation that minimizes the deformation energy E associated with the Laplacian Δ of the initial surface X from the generated deformation group of the initial surface X.
4. The initial curved surface X is subjected to the deformation obtained by the minimum energy deformation searching means 14 to generate a blended curved surface, and the curved surface display device 2
2 and the blended curved surface generating means 15 for outputting to 2. In the present embodiment, the curved surface modeler 2
One that supports a bicubic NURBS surface as 1 (for example, reference: "3D shape modeling for CAD / CAM, CG", PIXEL, No. 90, March 1990, 98-1).
See page 62). In such a surface modeler,
By specifying the control points, their weights, and knot vectors, which will be described later, with the mouse or keyboard, the curved surfaces A, B
An initial curved surface X that continuously connects the curved surfaces A and B is configured.

Next, the operation of this embodiment will be described. First,
The relationship among the curved surfaces A, B, and X and the deformation of the initial curved surface X in this embodiment will be described with reference to FIG.

The two curved surfaces A and B to be connected to each other are both bicubic NURBS curved surfaces, and the initial curved surface X is these two curved surfaces.
The curved surfaces A and B are connected continuously (but not smoothly). The initial surface X is also a bicubic NURBS surface,
It is appropriately designated according to the two curved surfaces A and B that are desired to be connected to each other. Here, the intersection line L _AX between the curved surface A and the initial curved surface X, the curved surface B
The intersection line L _BX between the initial curved surface X and the initial curved surface X is an open curve, but the present invention can also be applied to the case where both intersection lines are closed curves. The initial curved surface X can be a blended curved surface represented by a dotted line in the drawing by appropriately deforming it in the direction of the outer normal (the direction of the arrow in the drawing).

The curved surface modeler 21 converts the data of the curved surfaces A, B and X, which are bicubic NURBS curved surfaces, into knot vectors (t ₀ ⁽¹⁾ , ..., T _{m + 1} ⁽¹⁾ ), respectively. (T ₀ ⁽²⁾ ,
..., tn ₊₁ ⁽²⁾ ), the control points {P _{i, j} } and their weights {w _{i, j} }, (0 ≦ i ≦ m, 0 ≦ j ≦ n) in the blend surface generator 10. Output. At this time, generally, the bicubic NURBS surface S is parametrically expressed as an image of the next mapping as follows.

[0016]

[Equation 2]Where M_{i, 4}, N_{j, 4}Are knot vectors
(T₀ ⁽¹⁾, ..., t_{m + 1} ⁽¹⁾), (T ₀ ⁽²⁾, ..., tn₊₁ ⁽²⁾)
Is a B-spline function of the fourth order with respect to.

When the blended curved surface generator 10 receives the above data from the curved surface modeler 21, the parametric expression generating means 11 performs parameter conversion on the input data and expresses each input data as an image of the next mapping. Gives a parametric expression that

[0018]

[Equation 3] However,

[0019]

[Equation 4] Is. Further, this parametric expression generating means 11
Is the parametric representation of the unit exterior vector field of each surface A, B, X from this parametric representation.

[0020]

[Equation 5] To generate.

Each parametric expression obtained by the parametric expression generation means 11 is output to the modified group generation means 12 and the Laplacian calculation means 13. When the two or more integers m ₁ and m _{2 of} 4 or more are given, the transformation group generating means 12 uses each parametric expression regarding the curved surfaces A, B, and X obtained by the parametric expression generating means 11 to generate an initial curved surface. A family of transformations of X into a blended curved surface between curved surfaces A and B is generated. The transformation family is defined as a transformation family (set) as described above. This variant is a curved surface A, B
When the intersection lines L _AX and L _BX between the curved surface X and the curved surface X are both open curves (case O) and closed curves (case C)
It depends on the

[0022]

[Outer 1] It is obtained as an image of the map to.

[0023]

[Equation 6] Here, each λ _ij is an arbitrary real number. In case O,

[0024]

[Equation 7] And in case C,

[0025]

[Equation 8] Is. Also,

[0026]

[Equation 9] Is. However,

[0027]

[Equation 10] By the way, for μ = 1,2 and arbitrary integers i, j,

[0028]

[Equation 11] However,

[0029]

[Equation 12] And an integer j and u ₂ ε [a such that 0 ≦ j ≦ m ₂ −1.
₂ , b ₂ ],

[0030]

[Equation 13] Is.

The deformation group of the initial curved surface X thus obtained is output to the minimum energy searching means 14 and the blend curved surface generating means 15.

On the other hand, the Laplacian calculating means 13 calculates the Laplacian Δ and the area element dA of the initial curved surface X using the parametric expression obtained by the parametric expression generating means 11. Laplacian Δ is [a ₁ , b ₁ ]
It is the following differential operator for an arbitrary C ² function φ on × [a ₂ , b ₂ ]. This function φ represents the deformation of the initial curved surface X.

[0033]

[Equation 14] Here, for integers i, j such that 1 ≦ i, j ≦ 2,

[0034]

[Equation 15] Is. However, (g _ij ) is 2 × 2 having {g _ij } as a component
Represents a matrix, and g ^ij represents the ij-th component of a 2 × 2 matrix (g _ij ) ^-1 . The area element dA is given by the following equation.

[0035]

[Equation 16] The Laplacian Δ and the area element d thus obtained
A is output to the minimum energy deformation search means 14.

Subsequently, the minimum energy deformation search means 14
Searches for a deformation that minimizes the deformation energy E among the deformation groups generated by the deformation group generation means 12. The deformation of the initial curved surface X in the present embodiment deforms the initial curved surface X in the outer normal direction,

[0037]

[Equation 17] Is a variation of the type given by (where each λ _ij is a real number).
Of the deformation groups represented in this way, the deformation energy E
Although the blended curved surface is obtained by the deformation that minimizes, the minimum energy deformation search means 14 uses the deformation energy E
The deformation that minimizes the deformation energy E is determined by calculating the set of real numbers {λ _ij } that gives the deformation that minimizes. This set {λ _ij } is

[0038]

[Equation 18] Given by. here,

[0039]

[Formula 19] Is. The obtained set of real numbers {λ _ij } is transmitted to the blend curved surface generating means 15.

Finally, the blended curved surface generating means 15 generates the blended curved surface to be obtained, and the generated blended curved surface is output to the curved surface display device 22 connected to the blended curved surface generating device 10. As described above, the minimum deformation energy searching means 14 outputs a set of real numbers {λ _ij } related to the deformation that minimizes the deformation energy E. Therefore, the deformations generated by the set of real numbers {λ _ij } and the deformation family generation means. From the group, the blended curved surface generating means 15 determines the blended curved surface to be obtained.

[0041]

[Outside 2] Mapping to

[0042]

[Equation 20] Generated as an image of.

By the above operation, given two curved surfaces A,
From B and the initial curved surface X, a blended curved surface that smoothly connects the two curved surfaces A and B is generated and output. Since this blended curved surface has the smallest deformation energy E associated with the Laplacian of the initial curved surface X among the blended curved surfaces obtained by deforming the initial curved surface X, the global characteristic of the shape of the initial curved surface X is determined. Hold and the initial curved surface X
This corresponds to the deformation when the is regarded as a homogeneous elastic film.

[0044]

As described above, the present invention is based on the deformation group generating means for generating the deformation group of the initial curved surface, the Laplacian calculating means for calculating the Laplacian of the initial curved surface, and the result calculated by the Laplacian calculating means. By providing a minimum energy deformation search means for searching for a deformation that minimizes the deformation energy associated with Laplacian in the generated deformation family, the generated blended curved surface retains the global features having the shape of the initial curved surface. In addition, there is an effect that an intuitive understanding can be easily performed because the deformation is accompanied by a physical interpretation.

[Brief description of drawings]

FIG. 1 is a block diagram showing the configuration of a blended curved surface generator according to an embodiment of the present invention.

FIG. 2 is a diagram illustrating a modified example from an initial curved surface to a blended curved surface.

[Explanation of symbols]

10 Blended Curved Surface Generator 11 Parametric Expression Generator 12 Modified Group Generator 13 Laplacian Calculator 14 Minimum Energy Deformation Searcher 15 Blended Curve Generator 21 Curved Modeler 22 Curved Surface Display A, B Curved L _AX , L _BX Intersection X Initial curved surface

Claims (1)

[Claims] 1. An initial curved surface for continuously connecting between the given two curved surfaces is specified for the given two curved surfaces, and the given two curved surfaces are deformed by deforming the initial curved surfaces. In a blended curved surface generator that generates a blended curved surface that smoothly connects between the two, a deformation group generation unit that generates a deformation group of the initial curved surface when the blended curved surface is generated by deforming the initial curved surface in the normal direction thereof. A Laplacian calculation unit that calculates the Laplacian of the initial curved surface; and a deformation energy that accompanies the Laplacian among the deformation groups generated by the deformation group generation unit based on the result calculated by the Laplacian calculation unit. And a minimum energy deformation search means for searching for a deformation, and the deformation searched by the minimum energy deformation search means for the first time. Blend curved generating apparatus characterized by having a blended surface generation means for generating a blended curved by applying a curved surface.