Five-point difference format-based interpolation curved surface generation method

Abstract

The invention discloses an interpolation curved surface generation method based on a five-point difference format, and relates to the technical field of three-dimensional modeling and curved surface generation, four NURBS curves are input, a parameter domain Omega is equally divided along a u direction and a v direction, and (n + 1) * (m + 1) grid points are obtained through discretization on the parameter domain Omega, and each grid point is a control point {Pi, ji = 0, 1, 2,..., n, j = 0, 1, 2,..., m} of an interpolation curved surface; wherein P0, j, Pn, j, Pi, 0, Pi, m are control points on the four NURBS curves, namely known control points; (Pi, ji = 1, 2,..., n-1, j = 1, 2,..., m-1) is an internal control point of the interpolation curved surface, namely an unknown control point; a five-point difference format is used for representing an internal control point and recording the internal control point as a vector P, solving of the internal control point is converted into solving AP = b, QR decomposition is carried out on a matrix A to make A = QR, solving of the internal control point is converted into solving RP = QTb, then solving is carried out to obtain the vector P, and an interpolation curved surface S (u, v) can be generated according to the control point. The invention provides a curved surface modeling method for interpolating a given boundary NURBS (Non-Uniform Rational B-Spline) curve based on a partial differential equation five-point difference format. The curved surface modeling method can be adaptive to the condition that the given boundary curve is complex to generate a regular curved surface interpolated to the boundary curve.

Description

Technical Field

[0001] The invention relates to the technical field of three-dimensional modeling and surface generation, in particular to an interpolation surface generation method based on a five-point difference format. Background Art

[0002] Given four boundary curves, generating a surface interpolated on these four boundary curves is a commonly used surface modeling method in CAD modeling, engineering design, computer graphics, medical image processing and terrain modeling.

[0003] The existing Coons surface generation technology is a common surface modeling method of four-boundary curve interpolation, which can be used to create smooth and continuous surfaces. Coons surface generates the internal parameterization of the surface by interpolating between adjacent boundary curves. Coons surface has the advantages of local control and wide applicability, but it also has some limitations and challenges. For example, when the input boundary curve geometry is complex, the complexity of the Coons surface is limited by the structural form of the surface itself; when the boundary curves are close, singularities are prone to occur in the generated Coons surface.

[0004] Since the 1980s, researchers in geometric design have begun to study geometric design based on partial differential equations. Bloor and Wilson generated hybrid surfaces and N-gon hole-filling algorithms based on biharmonic equations on rectangular computational domains. Taubin discretized the Laplace-Beltrami operator on surfaces and performed mesh geometry processing. Kobbelt constructed an interpolation subdivision format by discretizing the approximate Laplace operator. Another widely used partial differential equation method is called surface flow technology. Kobbelt uses surface diffusion flow to solve surface hybrids and free-form surface fitting. In 2006, Professor Xu Guoliang and Professor Pan Qing, predecessors of computational geometry in my country, used second-order, fourth-order and sixth-order mean curvature flows to deal with surface modeling problems with G0, G1 and G2 geometric continuity conditions. In 2012, Professor Xu Guoliang introduced the geometric partial differential equation method in computational geometry in detail in his monograph "Geometric Partial Differential Equation Methods in Computational Geometry".

[0005] The partial differential equation method in the early geometric design is to study the algorithm by taking the surface itself as the solution of the partial differential equation. On the one hand, due to the physical properties of the partial differential equation, the surface designed by this method has good modeling characteristics. On the other hand, when dealing with the surface design that satisfies the geometric continuity splicing, due to the change of the boundary conditions of the differential equation, different differential equations need to be selected according to different geometric continuity requirements, which is inconvenient to handle. Summary of the invention

[0006] In order to overcome the defects in the above-mentioned prior art, the present invention provides an interpolation surface generation method based on a five-point difference format, which can adaptively generate a regular surface interpolated on the boundary curve in the case of a complex given boundary curve.

[0007] To achieve the above object, the present invention adopts the following technical solutions, including:

[0008] Interpolation surface generation method based on five-point difference format,

[0009] Input four NURBS curves, including two NURBS curves along the v direction, namely C1(v) and C2(v), and two NURBS curves along the u direction, namely C3(u) and C4(u). The parameter domain (u,v)∈Ω formed by the four NURBS curves;

[0010] Divide the parameter domain Ω into n+1 and m+1 equal parts along the u direction and v direction respectively, let h1 and h2 be the equal division steps in the u direction and v direction respectively, and discretize (n+1)×(m+1) grid points on the parameter domain Ω, each grid point is the control point of the interpolation surface S(u,v);

[0011] The four NURBS curves input are as follows:

[0012]

[0013] Among them, {P i,j |i=0,1,2,…,n;j=0,1,2,…,m} is a point in three-dimensional space, which is the control point of the interpolation surface S(u,v); where P 0,j , P n,j , P i,0 , P i,m are the control points on the four input NURBS curves, and are the known control points of the interpolation surface S(u,v); {P i,j |i=1,2,…,n-1;j=1,2,…,m-1} are the internal control points of the interpolation surface S(u,v), i.e., the unknown control points;

[0014] Internal locus of control i,j The five-point difference format of |i=1,2,…,n-1;j=1,2,…,m-1} is as follows:

[0015]

[0016] Let vector P = (P 1,1 ,P 1,2 ,…,P 1,m-1 ,P 2,1 ,P 2,2 ,…,P 2,m-1 ,…,Pn-1,1 ,P n-1,2 ,…,P n-1,m-1 ) T Under the boundary conditions of the four NURBS curves, the solution of the internal control point is transformed into the solution of AP = b, and the matrix A is decomposed by QR so that A = QR, and the solution of the internal control point is transformed into the solution of RP = Q T b, and then solve the vector P to obtain the internal control point {P i,j |i=1,2,…,n-1; j=1,2,…,m-1};

[0017] According to the control points {P i,j |i=0,1,2,…,n;j=0,1,2,…,m}, generate the interpolation surface S(u,v).

[0018] Preferably, for hybrid surface generation, two given surfaces S1(u,v) and S2(u,v) are spliced, and a hybrid surface is generated between the surfaces S1(u,v) and S2(u,v), so that the surfaces S1(u,v) and S2(u,v) smoothly transition at the connection to form an overall continuous surface;

[0019] Given the surfaces S1(u,v) and S2(u,v) to be joined:

[0020]

[0021] According to G x Smoothness requirement: the first x control points on the blended surface that are adjacent to surfaces S1(u,v) and S2(u,v) are determined by the control points on surfaces S1(u,v) and S2(u,v);

[0022] The remaining internal control points on the blended surface are obtained using the five-point difference format to generate the blended surface.

[0023] Preferably, it is used for lofting surface generation, and a spline surface is generated using a given B-spline / NURBS curve;

[0024] For two adjacent curves C w (v) and C w+1 (v):

[0025]

[0026] The curve C w (v) and C w+1 (v) As the boundary conditions of the five-point difference format, the internal control points of the spline surface are obtained according to the five-point difference format to generate the spline surface.

[0027] Preferably, for generating a swept surface, the swept surface is generated using the contour line C(v) and the ridge line T(u) of the path along which the contour line moves;

[0028] The contour line C(v) is:

[0029]

[0030] The path ridge T(u) is:

[0031]

[0032] Define C5(v):

[0033]

[0034] Where Q0 = M n , Q i+1 -Q i =P i+1 -P i , i=0,1,…,m-1;

[0035] Define C6(u):

[0036]

[0037] Where N0 = P n , N m =Q n , N i+1 -N i =M i+1 -M i , i=0,1,…,n-2;

[0038] The four curves C(v), T(u), C5(v) and C6(u) are used as the four boundary curves of the swept surface. The internal control points of the swept surface are obtained according to the five-point difference format to generate the swept surface.

[0039] The advantages of the present invention are:

[0040] (1) The present invention proposes a surface modeling method for interpolating a given boundary NURBS curve based on a five-point difference format of a partial differential equation. The method is a modeling design method that uses a NURBS surface control network as a solution to a harmonic equation. The method can adaptively generate a regular surface interpolated on the boundary curve in response to the complexity of the given boundary curve.

[0041] (2) The present invention uses a unified algorithm to generate surfaces that meet different geometric continuity requirements.

[0042] (3) The present invention can be used for classic surface modeling functions such as blending, lofting, and generation of swept surfaces in the geometric kernel of industrial software. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 Input the relative positions of four NURBS curves.

[0044] Figure 2 Schematic diagram of the relationship between geometric smooth splicing and control points.

[0045] Figure 3 To generate a G based on the five-point difference format 0 Example of generating geometrically smooth spliced ​​and blended surfaces.

[0046] Figure 4 To generate a G based on the five-point difference format 1 Example of generating geometrically smooth spliced ​​and blended surfaces.

[0047] Figure 5 To generate a G based on the five-point difference format 2 Example of generating geometrically smooth spliced ​​and blended surfaces.

[0048] Figure 6 Schematic diagram of lofted surface.

[0049] Figure 7 Generate an example for a lofted surface.

[0050] Figure 8 Create an example for a swept surface. DETAILED DESCRIPTION

[0051] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0052] Before describing the method of the present invention, definitions of terms related to the present invention are given first. The detailed definitions of these terms can also be found in the prior art.

[0053] Definition 1: Let P have n+1 control points i The p-order NURBS curve (i=0,1,…,n) is:

[0054]

[0055] in,

[0056]

[0057] N i,p (u) is the vector defined at the node k u The i-th B-spline basis function on u =(u0,u1,…,u n ,…,u n+p+1 ), u∈[u0,u n+p+1 ].

[0058] Definition 2: Let there be (m+1)×(n+1) control points {P i,j The (p,q)-order NURBS surface with |i=0,1,2,…,n;j=0,1,2,…,m} is:

[0059]

[0060] in,

[0061]

[0062] Defined in On, where k u =(u0,u1,…,u n ,…,u n+p+1 ), k v =(v0,v1,…,v m ,…,v m+q+1 ), (u,v)∈[u0,u n+p+1 ]×[v0,v m+q+1 ].

[0063] Definition 3: Two NURBS surfaces are called and The connection along the v direction is G 0 Yes, if Right now

[0064]

[0065] Where S i (u,v) is defined in On, i=0,1, k v =(v0,v1,…,v m ,…,v m+q+1 ).

[0066] Definition 4: The connection between two NURBS surfaces S1(u,v) and S0(u,v) along the v direction as defined in Definition 2 is called G 1 If their connection is G 0 , and the first-order partial derivatives of the u variable at the boundary are in the same direction, that is,

[0067]

[0068] Right now

[0069]

[0070] Wherein, α>0 is an adjustable positive constant.

[0071] Definition 5: The junction of two NURBS surfaces S1(u,v) and S0(u,v) along the v direction defined in Definition 2 is called G 2 If their connection is G 1 , and the second-order partial derivatives of the u variable at the boundary are in the same direction, that is,

[0072]

[0073] Right now

[0074]

[0075] Wherein, β>0 is an adjustable positive constant.

[0076] Example 1: Interpolation surface generation based on five-point difference format

[0077] The interpolation surface generation method based on the five-point difference format of the present invention is specifically as follows:

[0078] Input four NURBS curves, including two NURBS curves along the v direction, namely C1(v) and C2(v), and two NURBS curves along the u direction, namely C3(u) and C4(u). The parameter domain formed by the four NURBS curves is (u,v)∈Ω=[a,b]×[c,d];

[0079] Considering that the surface output by the present invention is a NURBS surface, a discrete format is used to solve the harmonic equations that satisfy the first type of boundary conditions. For simplicity, the parameter domain Ω is divided into n+1 and m+1 equal parts along the u direction and the v direction, respectively, and h1 and h2 are recorded as the equal division steps in the u direction and the v direction, respectively. The control points of the interpolation surface S(u,v) are used as discrete variables to solve the harmonic equations, so that the internal control points of the interpolation surface S(u,v) can be output, that is, at this time, the discrete grid of the parameter domain Ω has (n+1)×(m+1) grid points, each grid point is a control point of the interpolation surface S(u,v), and the interpolation surface S(u,v) with four given boundaries is generated.

[0080] Divide the parameter domain Ω into n+1 and m+1 equal parts along the u direction and v direction respectively, let h1 and h2 be the equal division steps in the u direction and v direction respectively, and discretize (n+1)×(m+1) grid points on the parameter domain Ω, each grid point is the control point of the interpolation surface S(u,v);

[0081] The four NURBS curves input are

[0082]

[0083] Among them, {P i,j |i=0,1,2,…,n;j=0,1,2,…,m} is a point in three-dimensional space, which is the control point of the interpolation surface S(u,v); where P 0,j , P n,j , P i,0 , P i,m are the control points on the four input NURBS curves, and are the known control points of the interpolation surface S(u,v); {P i,j |i=1,2,…,n-1;j=1,2,…,m-1} are the internal control points of the interpolation surface S(u,v), that is, the unknown control points. In this embodiment, the relative positions of the four NURBS curves are as follows: Figure 1 As shown, a NURBS surface with the four NURBS curves as boundary curves is output, namely, an interpolation surface S(u,v).

[0084] With discrete solution of internal control points {P i,j The five-point difference format of |i=1,2,…,n-1;j=1,2,…,m-1} is as follows:

[0085]

[0086] If the same step size is used in the u and v directions, that is, h1 = h2, the five-point difference format is equivalent to:

[0087]

[0088] Let vector P = (P 1,1 ,P 1,2 ,…,P 1,m-1 ,P 2,1 ,P 2,2 ,…,P 2,m-1 ,…,P n-1,1 ,P n-1,2 ,…,P n-1,m-1 ) T Under the boundary conditions of the four NURBS curves, the solution of the internal control points is transformed into solving the linear equation system AP = b, and the matrix A is decomposed by QR so that A = QR, and the solution of the internal control points is transformed into solving RP = QT b, and then solve the vector P to obtain the internal control point {P i,j |i=1,2,…,n-1;j=1,2,…,m-1};According to the control points {P i,j |i=0,1,2,…,n;j=0,1,2,…,m}, generate the interpolation surface S(u,v).

[0089] For example: Consider the case of n = 4, m = 3, the internal control points form a vector P = (P 1,1 ,P 1,2 ,P 2,1 ,P 2,2 ,P 3,1 ,P 3,2 ) T , then AP = b is:

[0090]

[0091] Example 2: Blended Surface Generation

[0092] Surface blending refers to connecting two or more unconnected NURBS surfaces together through transition surfaces according to certain smoothness requirements, so that they transition smoothly at the connection to form an overall continuous surface.

[0093] Given the surfaces S1(u,v) and S2(u,v) to be joined, such as Figure 2 As shown,

[0094]

[0095] According to G x Smoothness requirement: the first x rows of control points on the mixed surface S0(u,v) that are adjacent to the surfaces S1(u,v) and S2(u,v) are determined by the control points on the surfaces S1(u,v) and S2(u,v).

[0096] For example, in Figure 2 If S0(u,v) and S1(u,v) are in accordance with G 1 splicing, then for S0(u,v), P i,j ,i=0,1;j=0,1,…,m has been determined;

[0097] If S0(u,v) and S2(u,v) follow G 1 splicing, then for S0(u,v), P i,j ,i=n-2,n-1,n;j=0,1,…,m has been determined.

[0098] Therefore, the four curve control points as boundary conditions of the five-point difference format are as follows:

[0099] Control point of C1(v): P 1,j ,j=0,1,…,m;

[0100] Control point of C2(v): P n-2,j ,j=0,1,…,m;

[0101] Control point of C3(u): P i,0 ,i=0,1,…,n; where i=0,1,n-2,n-1,n has been determined, and other control points can be input by the user;

[0102] Control point of C4(u): P i,m ,i=0,1,…,n; where i=0,1,n-2,n-1,n has been determined, and other control points can be input by the user;

[0103] In this way, a hybrid surface that meets the splicing requirements can be generated. The algorithm steps are as follows:

[0104] Input: control points and node vectors of the surfaces to be joined S1(u,v) and S2(u,v), node vectors of the blended surface S0(u,v), and control points of the boundary curves C3(u) and C4(u) entered by the user.

[0105] Output: Control points of the blended surface S0(u,v).

[0106] Step 1, generate boundary conditions according to continuity requirements;

[0107] Step 2, call the five-point difference format to generate a linear equation system;

[0108] Step 3, solve the linear equations and generate internal control points.

[0109] For example: Given n=n1=n2=9, m=11.

[0110] Given three NURBS surfaces, the node vectors are

[0111] Among them, p=3, q=4.

[0112] Given two control points of boundary curves C3(u), C4(u):

[0113]

[0114] Given the control points of the surface S1(u,v) to be spliced:

[0115]

[0116] Given the control points of the surface S2(u,v) to be spliced:

[0117]

[0118] The output satisfies G at the splicing point 0 , G 1 , G 2 The mixed surface S0(u,v) is as follows Figure 3 , Figure 4 , Figure 5 shown.

[0119] Example 3: Lofted Surface Generation

[0120] Lofted surface generation refers to the generation of a spline surface passing through a given B-spline / NURBS curve. Figure 6 shown.

[0121] For two adjacent curves C w (v) and C w+1 (v) A five-point difference format is used to generate a NURBS surface, i.e., a spline surface, that passes through (interpolates) the two curves.

[0122] Suppose two adjacent curves C w (v) and C w+1 (v):

[0123]

[0124] Similar to the hybrid surface generation function, according to the curve C w (v) and C w+1 (v) Generate boundary conditions of the five-point difference format, and then generate internal control points of the surface according to the five-point difference format.

[0125] The steps for generating the lofted surface based on the five-point difference format are as follows:

[0126] Input: C w (v),w=1,2,…,W.

[0127] Output: After C1(v), C2(v), …, C W (v) NURBS surface.

[0128] Step 1, generate the spline surface S1(u,v) passing through C1(v) and C2(v) as the initial surface.

[0129] Step 2, according to the surface through C w (v) Smoothness requirements and S w-1 (u,v) control point, generate through C w(v) and C w+1 (v) NURBS surface S w (u,v), w=2,3,…,W-1.

[0130] For example: given n=15, m=14.

[0131] Given that the node vectors of all NURBS surfaces are

[0132] Among them, p=4, q=5.

[0133] Given a boundary curve B 2k-1 (u), B 2k The control points of (u) are:

[0134]

[0135]

[0136] Given interpolation curves C1(v), C2(v), C3(v), C4(v). Its control points k=1,2,3,4,j=0,1,…,m is

[0137]

[0138]

[0139]

[0140]

[0141] Generate a lofted surface interpolated on these four curves.

[0142] like Figure 7 As shown, the surfaces S1(u,v) and S2(u,v) both satisfy G at the joints. 0 Condition, surfaces S2(u,v) and S3(u,v) both satisfy G at the joints 2 condition.

[0143] Example 4: Sweep Surface Generation

[0144] As an efficient tool for creating complex three-dimensional models, sweeping occupies an important position in computer-aided design. Sweeping mainly consists of two parts, one is the contour line C(v) that describes the initial shape, and the other is the path ridge line T(u) along which the contour line moves. The classic swept surface will generate a surface with self-intersection, degeneration, and discontinuity. Therefore, when actually generating a swept surface, an approximate method is used to generate the swept surface. The swept surface generated here makes each curve passing through the ridge line similar to the given contour line.

[0145] The control points of the surface in the present invention satisfy the harmonic equation. According to the properties of the harmonic equation, the internal control points will be evenly adjusted according to the distribution of the control points of the boundary curve. Therefore, the control points of the surface section at both ends of the ridge can be uniformly distributed by translating the contour line to the end point of the ridge line. The internal control point distribution is controlled by the harmonic equation to be as consistent as possible with the control point distribution of the boundary. At the same time, the control network can be avoided from self-intersection and degeneration, thereby generating a regular approximate swept surface.

[0146] Let the contour be:

[0147]

[0148] The ridgeline is:

[0149]

[0150] In order to make the generated surface have a sweeping effect, C5(v) can be defined as:

[0151]

[0152] Where Q0 = M n , Q i+1 -Q i =P i+1 -P i , i=0,1,…,m-1;

[0153] In addition, define C6(u):

[0154]

[0155] Where N0 = P n , N m =Q n , N i+1 -N i =M i+1 -M i , i=0,1,…,n-2;

[0156] The four curves C(v), T(u), C5(v) and C6(u) are used as the four boundary curves of the swept surface. The internal control points of the swept surface are obtained according to the five-point difference format to generate the swept surface.

[0157] The steps for generating the swept surface based on the five-point difference method are as follows:

[0158] Input: contour line C(v) and ridge line T(u).

[0159] Output: A swept surface with C(v) as contour and T(u) as ridge.

[0160] Step 1: Generate the above C5(v) and C6(u).

[0161] Step 2: Generate internal control points using C(v), T(u), C5(v), and C6(u) as boundary conditions in a five-point format, and then output the surface.

[0162] For example: given n=10, m=11.

[0163] The node vectors of the three NURBS surfaces are

[0164] Among them, p=3, q=3.

[0165] Given the control point P of the contour line C(v) j ,j=0,1,…,m, control point T of ridge line T(u) i ,i=0,1,…,n, generating two boundary curves C5(v) and C6(u).

[0166]

[0167]

[0168] After the boundary conditions are given, the following is generated: Figure 8 The swept surface shown.

[0169] The above are only preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. The interpolation surface generation method based on the five-point difference format is characterized by: Input four NURBS curves, including two NURBS curves along the v direction, namely C1(v) and C2(v), and two NURBS curves along the u direction, namely C3(u) and C4(u). The parameter domain (u,v)∈Ω formed by the four NURBS curves; Divide the parameter domain Ω into n+1 and m+1 equal parts along the u direction and v direction respectively, let h1 and h2 be the equal division steps in the u direction and v direction respectively, and discretize (n+1)×(m+1) grid points on the parameter domain Ω, each grid point is the control point of the interpolation surface S(u,v); The four NURBS curves input are as follows: Among them, {P i,j |i=0,1,2,…,n;j=0,1,2,…,m} is a point in three-dimensional space, which is the control point of the interpolation surface S(u,v); where P 0,j , P n,j , P i,0 , P i,m are the control points on the four input NURBS curves, and are the known control points of the interpolation surface S(u,v); {P i,j |i=1,2,…,n-1;j=1,2,…,m-1} are the internal control points of the interpolation surface S(u,v), i.e., the unknown control points; Internal locus of control i,j The five-point difference format of |i=1,2,…,n-1;j=1,2,…,m-1} is as follows: Let vector P = (P 1,1 ,P 1,2 ,…,P 1,m-1 ,P 2,1 ,P 2,2 ,…,P 2,m-1 ,…,P n-1,1 ,P n-1,2 ,…,P n-1,m-1 ) T Under the boundary conditions of the four NURBS curves, the solution of the internal control point is transformed into the solution of AP = b, and the matrix A is decomposed by QR so that A = QR, and the solution of the internal control point is transformed into the solution of RP = Q T b, and then solve the vector P to obtain the internal control point {P i,j |i=1,2,…,n-1; j=1,2,…,m-1}; According to the control points {P i,j |i=0,1,2,…,n;j=0,1,2,…,m}, generate the interpolation surface S(u,v).

2. The interpolation surface generation method based on the five-point difference format according to claim 1 is characterized in that: Used for hybrid surface generation, splicing the given two surfaces S1(u,v) and S2(u,v), generating a hybrid surface between the surfaces S1(u,v) and S2(u,v), so that the surfaces S1(u,v) and S2(u,v) smoothly transition at the connection to form an overall continuous surface; Given the surfaces S1(u,v) and S2(u,v) to be joined: According to G x Smoothness requirement: the first x control points on the blended surface that are adjacent to surfaces S1(u,v) and S2(u,v) are determined by the control points on surfaces S1(u,v) and S2(u,v); The remaining internal control points on the blended surface are obtained using the five-point difference format to generate the blended surface.

3. The interpolation surface generation method based on the five-point difference format according to claim 1 is characterized in that: Used for lofting surface generation, using given B-spline / NURBS curve to generate spline surface; For two adjacent curves C w (v) and C w+1 (v): The curve C w (v) and C w+1 (v) As the boundary conditions of the five-point difference format, the internal control points of the spline surface are obtained according to the five-point difference format to generate the spline surface.

4. The interpolation surface generation method based on the five-point difference format according to claim 1 is characterized in that: Used for swept surface generation, using the contour line C(v) and the path ridge line T(u) of the contour line movement to generate the swept surface; The contour line C(v) is: The path ridge T(u) is: Define C5(v): Where Q0 = M n , Q i+1 -Q i =P i+1 -P i , i=0,1,…,m-1; Define C6(u): where N0 = P n , N m = Q n , N i+1 - N i = M i+1 - M i , i = 0, 1, …, n - 2; The four curves C(v), T(u), C5(v) and C6(u) are used as the four boundary curves of the swept surface. The internal control points of the swept surface are obtained according to the five-point difference format to generate the swept surface.

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