Shape-adjustable parametric curved surface generation method interpolated on boundary curve

Abstract

The invention discloses a method for generating a parametric curved surface interpolated on a boundary curve and with an adjustable shape, which comprises the following steps of: 1, generating a curved surface R1 by adopting a traditional Bernstein primary function in a u direction and adopting a generalized C-Bezier primary function with a shape parameter in a v direction; 2, a generalized C-Bezier primary function with shape parameters is adopted in the u direction, a traditional Bernstein primary function is adopted in the v direction, and a curved surface R2 is generated; 3, generalized C-Bezier primary functions with shape parameters are adopted in the u direction and the v direction, the v direction is interpolated to the v-direction boundary curve of the curved surface R1, the u direction is interpolated to the u-direction boundary curve of the curved surface R2, and a curved surface R3 is generated; and 4, accumulating the characteristics of the curved surfaces R1 and R2, and subtracting the boundary characteristics of the curved surface R3 to generate a parametric curved surface S. According to the method, the internal shape of the curved surface can be adjusted through independent shape parameters, and the construction efficiency and accuracy of the complex curved surface are improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of computer graphics, and specifically relates to a method for generating a parametric surface that interpolates a boundary curve and has an adjustable shape. Background Art

[0002] In computer graphics, computer-aided design (CAD), and engineering analysis, parametric surfaces are one of the main tools for describing the shape of objects. Existing methods for generating parametric surfaces include B-spline surfaces, Bézier surfaces, and NURBS (Non-Uniform Rational B-Spline) surfaces, which have been widely used in model construction, animation production, and industrial design. These traditional methods usually determine the shape of the entire surface through the layout of control points.

[0003] However, existing methods for generating parametric surfaces have limitations in surface shape control. The main reason is that both the surface boundary and the internal shape change with the variation of control points. Taking the Bézier-like surface as an example, either fine shape adjustment cannot be performed, or the surface boundary cannot be interpolated during the adjustment process, resulting in the modification of the internal shape of the surface directly affecting the shape of its characteristic lines. Usually, in order to achieve an ideal surface shape effect, a large number of control points often need to be adjusted complexly and repeatedly, making the surface generation process very cumbersome and seriously affecting the design efficiency.

[0004] Therefore, there is an urgent need to develop a new method for generating parametric surfaces to reduce the cumbersome adjustment of control points, simplify the surface design process, and thus improve the efficiency of generating complex surface shapes. Summary of the Invention

[0005] Aiming at the deficiencies of the existing technology, the purpose of the present invention is to provide a method for generating a parametric surface that interpolates a boundary curve and has an adjustable shape, which can adjust the internal shape of the surface through independent shape parameters, improving the construction efficiency and accuracy of complex surfaces.

[0006] To achieve the above purpose, the present invention adopts the following technical solutions: A method for generating a parametric surface that interpolates a boundary curve and has an adjustable shape, comprising the following steps: Step 1: Use the traditional Bernstein basis function in the direction, and use the generalized C-Bézier basis function with a shape parameter in the direction to generate the surface ; Step 2: Use the generalized C-Bézier basis function with a shape parameter in the direction, and use the traditional Bernstein basis function in the direction to generate the surface ; Step 3. In the direction and the direction, the generalized C-Bézier basis functions with shape parameters are adopted, and the direction interpolates the direction boundary curve of the surface the direction interpolates the direction boundary curve of the surface ; Step 4. Accumulate the characteristics of the surfaces and and subtract the boundary characteristics of the surface to generate the parametric surface .

[0007] Furthermore, the surface in the said Step 1 is expressed by the formula: In the formula, represents the traditional Bernstein basis function, represents the generalized C-Bézier basis function with shape parameters, is the shape parameter, =0,1,2,....,n。

[0008] Furthermore, the surface in the said Step 2 is expressed by the formula: In the formula, represents the traditional Bernstein basis function, represents the generalized C-Bézier basis function with shape parameters, is the shape parameter, =0,1,2,....,m。

[0009] Furthermore, the specific process of the said Step 3 is: Step 3.1. Set four traditional Bézier curves 、 、 and , and with their control points , , and and the shape parameter of the surface in the direction and the curved surface In the direction, the shape parameters , construct four generalized C-Bézier boundary curves 、 、 and , expressed as: Using the shape parameters and to adjust the positions of the control points of the boundary curves 、 、 and so that the boundary curves are geometrically continuous and conform to the expected boundary shape; Step 3.2: Adopt the construction method of the first type of Coons surface to generate the surface , expressed as: In the formula, , , and are the four corner points of the surface , and the values of and are in the range of [0, 1]; Step 3.3: Convert the surface into a surface that strictly interpolates the boundary curves 、 、 and , expressed as: In the formula, is the ruled surface formed by the boundary curves and , is the ruled surface formed by the boundary curves and , represents the corner point interpolation term, , and are respectively expressed as: = = 。

[0010] Furthermore, the parametric surface in step 4 is expressed by the formula: 。

[0011] Compared with the prior art, the present invention has the following technical effects: First, by combining the traditional Bernstein basis function and the generalized C-Bézier basis function with shape parameters, basic surfaces that simultaneously contain both basis functions are generated and . Since and adopt opposite basis functions in the direction and the direction, it is convenient to unify the characteristics of the subsequent surface and maintain the consistency of the surface in the direction. Then, based on the shape parameters of the surface in the direction and the shape parameters of the surface in the direction, boundary curves are constructed. Then, by using the construction method of the first type of Coons surface and introducing a function for boundary interpolation, a basic surface whose boundary always interpolates the boundary curve is generated , achieving precise control of the boundary shape of the surface. Then, the characteristics of the two basic surfaces and are accumulated, and the boundary contribution of the surface is subtracted to obtain a new parametric surface . In the process of constructing the parametric surface , the internal shape of the surface can be finely adjusted through independent shape parameters while keeping the boundary unchanged, without the need to adjust a large number of control points, improving the construction efficiency and accuracy of complex surfaces. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Figure 1 : Flow chart of the present invention; Figure 2 : Surface generated by the present invention ; Figure 3 : Surface generated by the present invention ; Figure 4 : Surface generated by the present invention ; Figure 5 : Parametric surface generated by the present invention ; Figure 6 : The number surface generated by the present invention The change of the mean curvature under different shape parameters. Specific implementation mode

[0013] The following further elaborates on the specific content of the present invention in conjunction with embodiments.

[0014] As Figure 1 shown, a method for generating a parametric surface that interpolates a boundary curve and has adjustable shape includes the following steps: Step 1, construct the surface In the direction, use the traditional Bernstein basis function to generate a Bézier curve. In the direction, use the generalized C-Bézier basis function with a shape parameter to generate a Bézier curve with a shape parameter, where the shape parameter is . According to control points , generate a surface as Figure 2 shown, expressed as: In the formula, where represents the traditional Bernstein basis function, represents the generalized C-Bézier basis function with a shape parameter, is the shape parameter, =0,1,2,....,n， Utilize the shape parameter to adjust the influence of the control points on the surface shape, thereby achieving flexible adjustment of local areas; By combining and two types of basis functions, the surface retains the advantages of the traditional Bézier surface in the direction and has more precise shape adjustment ability in the direction; Step 2, construct the surface In the direction, use the generalized C-Bézier basis function with a shape parameter to generate a Bézier curve with a shape parameter, where the shape parameter is . In the direction, use the traditional Bernstein basis function to generate a Bézier curve. According to control points , generate a surface as Figure 3 shown , expressed as: In the formula, represents the traditional Bernstein basis function, represents the generalized C-Bézier basis function with shape parameters, is the shape parameter, =0,1,2,....,m， Using the shape parameter to adjust the control points on the influence of the surface shape, so as to achieve flexible adjustment of the local area; From the surface and the surface of the expression, it can be seen that the surface and the surface both contain the traditional Bernstein basis function and the generalized C-Bézier basis function with shape parameters at the same time. The difference between the surface and the surface is that in the direction and the direction, opposite basis functions are used, which helps to unify the characteristics of the final surface and can prevent the inconsistency of the surface in the direction; Step 3. Construct the surface Step 3.1. Set four traditional Bézier curves connected end to end 、 、 and , with their control points , , and and the surface in the direction of the shape parameter and the surface in the direction of the shape parameter , construct four generalized C-Bézier boundary curves 、 、 and , expressed as: Since the boundary curves 、 、 and The control points come from the surface and the surface boundaries. Therefore, combining the shape parameters and adjust the boundary curves 、 、 and control point positions to make the boundary curves geometrically continuous and conform to the expected boundary shape; Step 3.2: Using the construction method of the first type of Coons surface, generate a surface as Figure 4 shown , expressed as: wherein, , , and are all four corner points of the surface , and take values in [0, 1]. The surface can be adjusted either by the boundary curve control points or by the shape parameters and of the generalized C-Bézier basis function; ; Step 3.3: To convert the surface into a surface that strictly interpolates the boundary curves 、 、 and , introduce a function for interpolation along the boundary and represent the surface as: wherein, is the ruled surface formed by the boundary curves and , is the ruled surface formed by the boundary curves and , represents the corner point interpolation term used to eliminate the repeated influence between the corner points, , and are respectively expressed as: = ​​​​​ = Step 4: Construct a parametric surface Accumulate the characteristics of the two base surfaces and and subtract the boundary contributions of the surface to generate a parametric surface as shown in Figure 5 , expressed as: By integrating the surfaces , and , not only the local characteristics of the surfaces and are retained, but also with the help of the boundary interpolation effect of the surface , the parametric surface is completely interpolated to the four boundary curves 、 、 and , achieving the purpose of adjusting the internal shape through the shape parameters and , making the construction of the parametric surface more flexible and efficient, and improving the construction efficiency and accuracy of complex parametric surfaces.

[0015] To verify the effectiveness of a parametric surface generation method proposed in this embodiment that interpolates to boundary curves and has adjustable shape, the parametric surface generated by it is respectively compared with the characteristics of several existing common surfaces. The results are shown in Table 1. It can be seen that in this embodiment, the surface shape can be adjusted only through the shape parameters, and without boundary derivatives, and the generated parametric surface has high smoothness and high degrees of freedom.

[0016] Table 1 Comparison results of the characteristics of the parametric surface and several existing surfaces Figure 6 (a)~(i) show the average curvature changes of the same base parametric surface under different shape parameters, where: 1), when the shape parameter in the -3, 1, -1, 1, and Shape parameter in the direction -1, 2, -3, the resulting parametric surface is as shown in Figure 6 (a); 2), When the shape parameter in the direction -3, 1, -1, 1, and the shape parameter in the direction 1, -2, -1, the resulting parametric surface is as shown in Figure 6 (b); 3), When the shape parameter in the direction -3, 1, -1, 1, and the shape parameter in the direction -2, 3, -3, the resulting parametric surface is as shown in Figure 6 (c); 4), In the shape parameter in the direction 4, -3, 4, -2, and the shape parameter in the direction -1, -3, -2, the resulting parametric surface is as shown in Figure 6 (d); 5), When the shape parameter in the direction 4, -3, 4, , and the shape parameter in the direction -2, 1, 1, the resulting parametric surface is as shown in Figure 6 (e); 6), When the shape parameter in the direction 4, -3, 4, -2, and the shape parameter in the direction 1, -3, when it is 1, the obtained parametric surface is as Figure 6 shown in (f); 7), in the shape parameter in the direction of -3, -3, 1, -2, and the shape parameter in the -1, -3, 1, when the obtained parametric surface is as Figure 6 shown in (g); 8), in the shape parameter in the direction of -1, 3, -1, 1, and the shape parameter in the 1, 2, 1, when the obtained parametric surface is as Figure 6 shown in (h); 9), in the shape parameter in the direction of 3, 2, 3, -3, and the shape parameter in the -2, -2, -3, when the obtained parametric surface is as Figure 6 shown in (i); It can be seen from Figure 6 (a) to (i) that by only adjusting the shape parameters in the direction of and the direction of, the internal shape of the parametric surface can be adjusted without affecting the shape of the boundary curve. Therefore, there is no need to adjust a large number of control points, improving the design efficiency.

Claims

1. A method for generating a parametric surface interpolated on a boundary curve and having an adjustable shape, characterized in that: The steps include: Step 1: Use the traditional Bernstein basis function in the u direction and the generalized C-Bézier basis function with shape parameters in the v direction to generate the surface R1; Step 2: Use the generalized C-Bézier basis function with shape parameters in the u direction and the traditional Bernstein basis function in the v direction to generate the surface R2; Step 3, generalized C-Bézier basis functions with shape parameters are used in both the u direction and the v direction, and the v direction is interpolated on the v direction boundary curve of the surface R1, and the u direction is interpolated on the u direction boundary curve of the surface R2, to generate the surface R3; Step 4: Add the characteristics of surfaces R1 and R2 and subtract the boundary characteristics of surface R3 to generate the parametric surface S.

2. The method for generating a parametric surface interpolated on a boundary curve and having an adjustable shape according to claim 1, characterized in that: The surface R1 in step 1 is expressed by the formula: In the formula, B i,m (u) represents the traditional Bernstein basis function, represents the generalized C-Bézier basis function with shape parameter, λ j is the shape parameter, j=0,1,2,....,n.

3. The method for generating a parametric surface interpolated on a boundary curve and having an adjustable shape according to claim 1, characterized in that: The surface R2 in step 2 is expressed by the formula: In the formula, B j,n (v) represents the traditional Bernstein basis function, B λi,m (u) represents the generalized C-Bézier basis function with shape parameter, λ i is the shape parameter, i=0,1,2,....,m.

4. The method for generating a parametric surface interpolated on a boundary curve and having an adjustable shape according to claim 1, characterized in that: The specific process of step 3 is as follows: Step 3.1: Set four traditional Bézier curves P connected end to end u0 , P u1 , P v0 and P v1 , with its control point P 0i , P ni , P j0 and P jm and the shape parameter λ of the surface R1 in the v direction j And the shape parameter λ of surface R2 in the u direction i , construct four generalized C-Bézier boundary curves P′ u0 , P′ u1 , P′ v0 and P′ v1 , expressed as: Using the shape parameter λ j and λ i Adjust the boundary curve P′ u0 , P′ u1 , P′ v0 and P′ v1 The control points are positioned so that the boundary curve is geometrically continuous and conforms to the expected boundary shape; Step 3.2: Use the construction method of the first type of Coons surface to generate surface R3, which is expressed as: Where P 00 , P 01 , P 10 and P 11 are the four corner points of surface R3, and the values ​​of u and v are [0, 1]; Step 3.3: Convert the surface R3 to be strictly interpolated on the boundary curve P′ u0 , P′ u1 , P′ v0 and P′ v1 The surface is represented as: R3(u,v)=f(u)+g(v)-h(u,v) Where f(u) is the boundary curve P′ u0 and P′ u1 Construct a ruled surface, g(v) is the boundary curve P′ v0 and P′ v1 The ruled surface is composed of h(u,v), which represents the corner interpolation term, and f(u), g(v) and h(u,v) are expressed as:

5. The method for generating a parametric surface interpolated on a boundary curve and having an adjustable shape according to claim 1, characterized in that: The parameter surface S in step 4 is expressed by the formula: S(u,v)=R1(u,v)+R2(u,v)-R3(u,v).

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