Curved surface structure lofting method based on shape interpolation and application thereof

Abstract

The invention discloses a curved surface lofting method based on shape interpolation. The method comprises the following steps: defining at least two single connected domains on a curved surface structure to be lofted; two single connected domains are selected to be conformally mapped to a complex plane to obtain corresponding mapping parameter domains, and affine stretching transformation between the mapping parameter domains corresponding to the two single connected domains is calculated; combining conformal mapping and affine stretching transformation of the two single connected domains to obtain Taihmiller space mapping, and calculating a stretching quotient; and calculating a corresponding Beltrami coefficient based on the expansion quotient so as to obtain an interpolation shape between the two single connected domains, and completing lofting processing of the to-be-lofted curved surface structure according to the interpolation shape. Continuous and uniform transition of the two Jordan areas can be achieved through an interpolation method, and rapid and stable lofting of the curved surface body is achieved.

Description

Technical Field

[0001] The present application relates to the technical field of structural optimization design, and in particular to a surface structure lofting method based on shape interpolation, a surface structure lofting device based on shape interpolation, an electronic device, and a computer-readable storage medium. Background Art

[0002] Lofting is the process of using multiple two-dimensional objects as the cross-section of the structure and sweeping them along a certain direction to form a complex three-dimensional object. It is often used in the generation of complex curved surfaces such as blades, and is a commonly used surface and curved surface body generation operation in mechanical structure design. Traditional lofting operations are mainly oriented to continuous parametric curves, such as NURBS curves, and curve masking calculations are realized through order increase and node refinement. The core idea of ​​surface lofting is to interpolate each intermediate surface / curve through a set of compatible intermediate structures. Although the calculation method of continuous parametric lofting surface generation is simple in principle, it is often difficult to find a smooth intermediate structure to achieve a robust and convenient lofting function. In addition, interactive design of lofting surfaces is a tedious iterative process that depends on the number, shape and position of the lofted surfaces. Since the interpolation method is embedded in the computer and cannot be controlled by the designer, it will take a lot of time to obtain the ideal lofted surface body configuration.

[0003] Therefore, how to achieve continuous and uniform transition between the two Jordan regions through interpolation methods and realize fast and stable surface lofting is an urgent problem to be solved. Summary of the invention

[0004] In order to overcome the defects of the above-mentioned prior art, an embodiment of the present invention provides a surface structure lofting method and application based on shape interpolation, which can realize continuous and uniform transition between two Jordan regions through an interpolation method, thereby realizing fast and stable surface body lofting.

[0005] On the one hand, an embodiment of the present invention proposes a surface structure lofting method based on shape interpolation, comprising: defining at least two simply connected domains on the surface structure to be lofted, and determining the corner points of each of the simply connected domains respectively to construct a topological quadrilateral corresponding to each of the simply connected domains; selecting any two of the simply connected domains, conformally mapping the topological quadrilateral corresponding to each of the selected simply connected domains into a rectangular area on a complex plane, obtaining a mapping parameter domain corresponding to each of the simply connected domains, and calculating the affine stretching transformation of the mapping parameter domain corresponding to the two simply connected domains; dividing the two simply connected domains into The conformal mapping corresponding to each other and the affine stretching transformation between the two simply connected domains are compounded to obtain a Teichmuller space mapping between the two simply connected domains, and a stretch quotient between the two simply connected domains is calculated based on the Teichmuller space mapping; an introduced variable between the two simply connected domains is determined based on the stretch quotient, and a Beltrami coefficient between the two simply connected domains is calculated based on the stretch quotient and the introduced variable; an interpolation shape between the two simply connected domains is obtained based on the Beltrami coefficient; and a lofting process of the surface structure to be lofted is completed according to the interpolation shape.

[0006] In one embodiment of the present invention, the calculating of the affine stretching transformation of the mapping parameter domain corresponding to the two simply connected domains includes: setting a harmonic mapping according to the conformal mapping from the topological quadrilateral to the rectangular region to satisfy: Among them, g m :S m →R m For the topological quadrilateral S m To the rectangular area R m The conformal mapping, g n :S n →R n For the topological quadrilateral S n To the rectangular area R n The conformal mapping, and are the boundaries of the two topological quadrilaterals, Δ is the Laplace operator defined on the complex plane; the stretching affine transformation h of the mapping parameter domain corresponding to the two simply connected domains is determined according to the harmonic mapping [m，n] .

[0007] In one embodiment of the present invention, the conformal mappings corresponding to the two simply connected domains and the affine stretching transformation between the two simply connected domains are combined to obtain the Teichmuller space mapping between the two simply connected domains, including: combining the stretching affine transformation h between the two simply connected domains [m，n] The conformal mappings g corresponding to the two simply connected domains respectively mand g n -1 Composite Mapping As described in Teichmuller space mapping.

[0008] In one embodiment of the present invention, determining an introduced variable between two simply connected domains based on the stretch quotient, and calculating the Beltrami coefficient between the two simply connected domains according to the stretch quotient and the introduced variable, include: making the stretch quotient change monotonically linearly to determine the introduced variable between the two simply connected domains, setting an interpolation function to establish a relationship in which the stretch quotient changes with changes in the introduced variable, and obtaining the Beltrami coefficient corresponding to the stretch quotient according to the interpolation function.

[0009] In one embodiment of the present invention, the simply connected domain includes multiple triangular facets in the form of triangulation, and the interpolation shape between two simply connected domains is obtained based on the Beltrami coefficients, including: obtaining the surface shape of each triangular facet after mapping according to the Beltrami coefficients, and assembling and stitching all the mapped surface shapes to obtain the interpolation shape between the two simply connected domains.

[0010] In one embodiment of the present invention, the method of obtaining the surface shape of each triangular face after mapping according to the Beltrami coefficient includes: obtaining the vertex set and connection relationship of the triangular face to determine the original triangulation mesh and the target triangulation mesh composed of the triangular facets on the two simply connected domains; using a piecewise linear function to approximate the mapping of the original triangulation mesh to the target triangulation mesh to obtain the linear change of each triangular facet in the original triangulation mesh mapped to the target triangulation mesh; and calculating the area of ​​each triangular facet and the Beltrami coefficient according to the linear change to construct the surface shape of each triangular facet after mapping.

[0011] In one embodiment of the present invention, the lofting processing of the surface structure to be lofted according to the interpolation shape includes: arranging the mapping parameter domain corresponding to the simply connected domain according to the surface structure to be lofted, and normalizing it in the direction of the distribution arrangement; performing linear interpolation on the normalized mapping parameter domain according to the interpolation shape to form a parameter hexahedron region; normalizing the parameter hexahedron region in the direction of linear interpolation to obtain a unit cube mapping structure corresponding to the surface structure to be lofted, and calculating the lofting configuration of the surface structure to be lofted according to the unit cube mapping structure.

[0012] On the other hand, an embodiment of the present invention further proposes a surface structure lofting device based on shape interpolation, comprising: a simply connected domain definition module, used to define at least two simply connected domains on the lofted surface structure, and determine the corner points of the simply connected domains to construct corresponding topological quadrilaterals; an affine stretching transformation module, used to select any two of the simply connected domains, conformally map the topological quadrilaterals corresponding to each of the selected simply connected domains into a rectangular area on the complex plane, obtain a mapping parameter domain corresponding to each of the simply connected domains, and calculate the affine stretching transformation of the mapping parameter domains corresponding to the two simply connected domains; a stretch quotient calculation module, used to respectively map the two simply connected domains to the topological quadrilaterals corresponding to the selected simply connected domains. The corresponding conformal mapping is compounded with the affine stretching transformation between the two simply connected domains to obtain a Teichmuller space mapping between the two simply connected domains, and the stretch quotient between the two simply connected domains is calculated based on the Teichmuller space mapping; an interpolation shape obtaining module is used to determine the introduced variables between the two simply connected domains based on the stretch quotient, and calculate the Beltrami coefficient between the two simply connected domains based on the stretch quotient and the introduced variables; an interpolation shape between the two simply connected domains is obtained based on the Beltrami coefficient; a lofting processing module is used to complete the lofting processing of the surface structure to be lofted according to the interpolation shape.

[0013] On the other hand, an embodiment of the present invention further proposes an electronic device, comprising: a memory and one or more processors connected to the memory, the memory storing a computer program, and the processor being used to execute the computer program to implement the surface structure lofting method based on shape interpolation as described in any one of the above embodiments.

[0014] On the other hand, an embodiment of the present invention further provides a computer-readable storage medium, wherein the computer-readable storage medium stores computer-executable instructions, and the computer-executable instructions are used to execute the surface structure lofting method based on shape interpolation as described in any one of the above embodiments.

[0015] It can be seen from the above that, compared with the prior art, the above embodiments of the present invention can have at least one or more of the following beneficial effects:

[0016] By combining mathematical concepts such as conformal geometry and Teichmuller space mapping with computational methods such as spline shape interpolation, a lofted surface body modeling and mesh generation method based on discrete surfaces is proposed. This method defines a simply connected domain on the lofted surface structure and determines the corner points to construct the corresponding topological quadrilateral. After conformal mapping to the complex plane, the affine stretch transformation of the mapping parameter domain is calculated, and the Teichmuller space mapping is obtained by compounding it with the conformal mapping of the simply connected domain and calculating the stretch quotient. The target interpolation function is set to establish the relationship between the stretch quotient and the variables introduced in the simply connected domain, and the corresponding Beltrami coefficients are calculated, thereby obtaining the design shape of the target interpolated simply connected domain. In this way, a smooth mapping is found to achieve a continuous and uniform transition between the Jordan regions of two planar curved edges, thereby achieving fast and stable surface body lofting. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] The drawings described herein are used to provide a further understanding of the present invention and constitute a part of the present invention. The exemplary embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation of the present invention. In the drawings:

[0018] Figure 1 A flow chart of a surface structure lofting method based on shape interpolation provided by an embodiment of the present invention;

[0019] Figure 2 A schematic diagram of the mapping principle between two adjacent lofted surfaces provided by an embodiment of the present invention;

[0020] Figure 3 A schematic diagram of a process for constructing a curved surface shape of a triangular patch mapping according to Beltrami coefficients provided in an embodiment of the present invention;

[0021] Figure 4 A schematic diagram of lofted surface modeling and parameterization provided in an embodiment of the present invention. DETAILED DESCRIPTION

[0022] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other. The present invention will be described below with reference to the accompanying drawings and in combination with the embodiments.

[0023] In order to enable ordinary technicians in the field to better understand the technical solution of the present invention, the technical solution in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments, and all should belong to the protection scope of the present invention.

[0024] It should be noted that the terms "first", "second", etc. in the specification and claims of the present invention and the above-mentioned drawings are applicable to distinguishing similar objects, and are not necessarily used to describe a specific order or sequence. It should be understood that the terms used in this way can be interchangeable where appropriate, so that the embodiments of the present invention described herein can be implemented in an order other than those illustrated or described herein. In addition. The terms "including" and "having" and any variations thereof are intended to cover non-exclusive inclusions, for example, a process, method, system, product or device that includes a series of steps or units is not necessarily limited to those steps or units that are clearly listed, but may include other steps or units that are not clearly listed or are proprietary to these processes, methods, products or devices.

[0025] It should also be noted that the division of multiple embodiments in the present invention is only for the convenience of description and should not constitute a special limitation. The features in various embodiments can be combined and referenced to each other without contradiction.

[0026] like Figure 1 As shown, the first embodiment of the present invention proposes a surface structure lofting method based on shape interpolation, for example, including: step S1, defining at least two simply connected domains on the surface structure to be lofted, and determining the corner points of each of the simply connected domains respectively to construct a topological quadrilateral corresponding to each of the simply connected domains; step S2, selecting any two of the simply connected domains, conformally mapping the topological quadrilateral corresponding to each of the selected simply connected domains to a rectangular area on a complex plane, obtaining a mapping parameter domain corresponding to each of the simply connected domains, and calculating the affine stretching transformation of the mapping parameter domains corresponding to the two simply connected domains; step S3, mapping the two simply connected domains to a rectangular area on a complex plane; and finally, calculating the affine stretching transformation of the mapping parameter domains corresponding to the two simply connected domains. The conformal mapping corresponding to the common domains and the affine stretching transformation between the two simply connected domains are compounded to obtain a Teichmuller space mapping between the two simply connected domains, and the stretch quotient between the two simply connected domains is calculated based on the Teichmuller space mapping; step S4, determining the introduced variables between the two simply connected domains based on the stretch quotient, and calculating the Beltrami coefficient between the two simply connected domains according to the stretch quotient and the introduced variables; obtaining an interpolation shape between the two simply connected domains based on the Beltrami coefficient; step S5, completing the lofting processing of the surface structure to be lofted according to the interpolation shape.

[0027] Specific, combined Figure 2 As shown, for example, define two plane curved edges Jordan region S on the surface structure to be lofted m and S n , S m and S n By selecting corner points to transform into topological quadrilaterals, two topological quadrilaterals are conformally mapped to the parameter domain R m and Rn Inside, the ratio of the vertical and horizontal coordinates of the rectangle is a m and a n That is S m and S n Further use of a m and a n Calculate from S m and S n The stretch quotient of the two parameter domains is obtained by [m，n] , establish a linear function of the stretch quotient with respect to parameter t, and calculate the Beltrami coefficient of each stretch quotient value corresponding to parameter t. Then, the interpolation shape is obtained according to the corresponding Beltrami coefficient. When parameter t is within the definition domain, the interpolation shape changes from S m and S n Continuous transition.

[0028] Furthermore, let the mapping corresponding to the parameter scalar t be The interpolation region corresponding to t is triangulated as The interpolation parameter t ranges from t∈[0,1]. m and S n Smooth shape interpolation between the two must meet the following conditions:

[0029] (1) When t∈[0,1], f t＝0 =Id and f t＝1 =f. For the interpolation of the mapping, it is necessary to satisfy that when t = 0, the mapping is to convert S m Mapped to itself, i.e. f t＝0 :S m →S m , when t = 1, the mapping is S m To S n The Teichmüller map, that is, f t＝1 =f:S m →S n .

[0030] (2) Point-wise bounded non-conformal distortion. Let point p be S m A point in the m , K(p) is the non-conformal distortion of point p, that is, the stretch quotient. And when the interpolation is obtained, the mapping at time t is When is the expansion quotient at this time, then Special note, if in f [m，n] :S m →S n In the special case that K(f)≡1 exists for any point, i.e., the mapping f is a conformal mapping, then for have Right now This also requires conformal mapping.

[0031] (3) Symmetry. Let f -1 :S m →S n is the inverse process of the mapping f, and It is from S n To S m is the interpolation result at time t calculated by the same method, then In other words, whether from S m To S n Or from S n To S m , the interpolation results for t are also one-to-one corresponding, and there is

[0032] (4) Smoothness and harmony. [0，1]×z It is C ∞ - continuous, for all z∈Ω, the derivative exists and is bounded. And for t∈[0,1], is a harmonic function, that is

[0033] Therefore, in step S1, for example, in the two Jordan regions S m and S n Find the corresponding counterclockwise corner points on and Constructed with S m and S n The topological quadrilateral (S m ,g,{p m，0 , p m，1 , p m，2 , p m，3}) and (S n ,g,{p n，0 , p n，1 , p n，2 , p n，3}).

[0034] In step S2, the two topological quadrilaterals are conformally mapped to R m and R n In the two plane rectangles, we get two rectangles R on the complex plane C. m =g m (S m ) and R n =g n (S n ):

[0035] g m :S m (p m，0 , p m，1 , p m，2 , p m，3 ) → R M (0, a m , a m +i,i)

[0036] g n :S n (p n，0 , p n，1 , p n，2 , p n，3 ) → R n (0, a n , a n +i,i)

[0037] In order to obtain continuous shape interpolation, we need to find a harmonic mapping that satisfies:

[0038]

[0039] in, and is the boundary of two topological quadrilaterals, Δ is the Laplacian operator defined on the complex plane. [m，n] In the above example, the ordinate of each point in the rectangle does not change, but the abscissa is transformed proportionally, that is:

[0040]

[0041] Therefore, if Figure 2 As shown, for the composite mapping Due to g m and g n -1 is a conformal mapping, and h [m，n] is harmonic, so f [m，n] is also harmonic. In addition, due to g m and g n -1 is a conformal mapping, then the mapping f [m，n] The conformal distortion and h [m，n] The conformal distortion of is the same. At the same time, it can be calculated The Beltrami coefficient is:

[0042]

[0043] This means:

[0044]

[0045] Similarly, we can get:

[0046]

[0047] Because h [m，n] : R m →R n is a stretching affine transformation, whose stretch quotient is K z (f) = κ mn , that is, a n / a m (a n >a m ) or a m / a n (a n ≤a m ), so f [m，n] The Beltrami coefficient can be calculated as:

[0048]

[0049] At this point, the Teichmüller distance of the mapping, that is, the maximum quasi-conformal expansion between the two parameter domains (further developed to the extremal mapping in the two plane curve domains), is uniquely determined. m and S n The Teichmüller distance d between them is:

[0050]

[0051] In step S3, for two conformal mappings, a stretched affine transformation (Teichmüller mapping) g m , g n -1 and h [m，n] The compound obtained from S m and S n Teichmüller Mapping:

[0052]

[0053] The mapping f can be calculated [m，n] After obtaining the expansion quotient K(f), we introduce the control variable t∈[0,1] into the two Jordan regional interpolation problems and select a suitable interpolation function to establish the relationship between the expansion quotient and the variable t. t (f), calculate the corresponding Beltrami coefficient and the relationship between the variable t μ t (f).

[0054] For two Jordan regions S on the complex planem and S n ,according to Figure 2 and the mapping relationship f given in step S1 [m，n] ,have

[0055] g m :S m (p m，0 , p m，1 , p m，2 , p m，3 ) → R m (0, a m , a m +i,i)

[0056] g n :S n (p n，0 , p n，1 , p n，2 , p n，3 ) → R n (0, a n , a n +i,i)

[0057] Computable mapping f [m，n] The expansion quotient is

[0058]

[0059] In step S4, when t∈[0,1], if the expansion quotient changes monotonically with the change of t, we can first assume that the change is linear, that is, K t (f) = (K(f)-1)·t+1, and according to K t (f) The Beltrami coefficient of the mapping at this time can be obtained as:

[0060]

[0061] Define a normalization coefficient To satisfy:

[0062]

[0063] And order Then the Beltrami coefficient at time t can be defined as:

[0064]

[0065] The Beltrami coefficient corresponding to any moment can be calculated based on the mapping and the t value at any moment.

[0066] Furthermore, for example, two Jordan regions S are input in triangulated formm and S n , S m To S n The mapping relationship For the convenience of calculation, for S m and S n The triangulation representation of S needs to unify the index order in its connection relationship, that is, for S m ={X m , F m} and S n ={X n , F n}, there is F m =F n , so S m and S n The connection relationship of the midpoints is uniformly represented by F. If the triangulation of the input surface does not correspond, the mapping relationship between the two surfaces can be used to represent the connection relationship between the midpoints. [m，n] S m The coordinate points on S are mapped to n On, and continue to use S m The triangulation index F on m .

[0067] The above steps provide a method to calculate the expansion quotient and Beltrami coefficient corresponding to any intermediate time t through the mapping relationship between two topological quadrilaterals. Theoretically, given the Beltrami coefficient μ t (f), there must be a smooth mapping f t Corresponding to it. However, in the actual operation process, it is impossible to obtain the expected piecewise linear mapping by directly solving the Beltrami equation. At the same time, since in the actual calculation of piecewise mapping, the Beltrami coefficient depends on the triangle face (i.e., the connection relationship F) rather than the coordinate point, the obtained equation cannot be solved for the discrete triangulation grid whose number of points is often only half of the number of faces. Therefore, for example, the solution of the mapping is divided into two steps: first, the shape of each triangle face after mapping is calculated under the specified stretch quotient and Beltrami coefficient, and then all the triangle faces are assembled and stitched together to obtain the target interpolation result.

[0068] Furthermore, using S m = {V m , F} to represent the region S m The triangulation grid on which X m is a set of triangle vertices, and F represents the connection relationship of the triangles. f=u+iv is the original mesh S m = {V m , F} to the target grid S n = {V n, F}, where the grid S m and S n The number of vertices is the same and they have the same connectivity F. Use piecewise linear functions to approximate the mapping f. Specifically, f is in each triangle F = [v i , v j , v k ] is a linear function:

[0069] f| F (x, y) = (a F x+b F y+p F )+i(c F x+d F y+q F )

[0070] where v i , v j and v k are the vertices of triangle F in counter-clockwise order.

[0071] like Figure 3 As shown, for each triangle, its Beltrami coefficient can be assumed to be μ t (F) = ρ t (F)+iτ t (F). If the point before mapping is v = g + ih, and the point after mapping is w = s + it, that is:

[0072]

[0073] Then for each triangle, according to the linear change, we can get:

[0074]

[0075]

[0076] Among them, there are:

[0077]

[0078]

[0079] Among them, A F is the area of ​​the triangle. At the same time, the Beltrami coefficient can be calculated on this triangle:

[0080]

[0081] Assume e k is the longest side of this triangle. For the corresponding local triangle, when assuming F′=[v i′ v j ′v k ′], and v i ′=(0,0),v j '=(1,0), then we can establish the equation:

[0082]

[0083] To solve for v′ kx and v′ kx .

[0084] Furthermore, after obtaining the shape of each triangle, all triangles need to be stitched together to form a new interpolated shape. In order to ensure that the shape of the triangle does not change, for example, each triangle is rotated and scaled proportionally so that all triangles can be seamlessly stitched together to form a new triangulated mesh.

[0085] Specifically, on the two-dimensional complex plane, the rotation and scaling matrices of a vector can be written as follows:

[0086]

[0087] Among them, c F′ and F′ At least one entry is non-zero. Using the mapping The expression of the local transformation on the triangle F′ can be obtained as:

[0088]

[0089] Where E represents the triangle boundary relationship, and E = {{i, j}, {j, k}, {k, i}}. Substituting the matrix form in the above expression into the expression, the optimized column form of each item in the coordinate transformation matrix can be obtained as follows:

[0090]

[0091] in,

[0092]

[0093] The optimization formula given by the above formula is a least squares problem, and its solution can be written as:

[0094]

[0095] remember are the vertices of the triangulated mesh after stitching, then It can be obtained by solving the following optimization problem:

[0096]

[0097]

[0098] The column form is about the variables The quadratic positive definite optimization problem can be written in classic matrix form, namely

[0099]

[0100] in, is the assembled coefficient matrix, each row of which corresponds to a summed term in the column form; the variable is a 2N×1 vector, where N is the number of coordinate points in the triangulation:

[0101]

[0102] b is a column vector of (M+2C)×1, where M is the number of terms to be summed in the first half of the formula, and C is the number of fixed points. From the matrix form, it can be seen that this optimization problem can be solved by the conjugate gradient method after some vertices are given. So far, the Teichmüller mapping between the Jordan regions on the two complex planes and the corresponding expansion quotient of each triangular facet are obtained, and the expansion quotient is interpolated and the corresponding Beltrami coefficient is calculated; further, according to the Beltrami coefficient interpolated from the intermediate state, the shape of each triangular facet is solved, and then all triangles are ARAP stitched, and the shape of the intermediate transition surface and the corresponding triangulation are obtained by solving a quadratic positive definite optimization problem.

[0103] Furthermore, lofted surface modeling and parameterization methods are as follows Figure 4 As shown in the figure, the lofted surface is conformally mapped to the rectangular parameter domain, and all the lofted surfaces are arranged according to the normalized T number distribution; secondly, the U-direction interpolation is completed by the linear interpolation method to form a parameter hexahedron region; then, the continuous parameters in the U direction are normalized to obtain the mapping of the lofted surface body to the unit cube; finally, the lofted surface body configuration is calculated according to this mapping.

[0104] In summary, the first embodiment of the present invention proposes a surface structure lofting method based on shape interpolation. By combining mathematical concepts such as conformal geometry and Teichmuller space mapping with calculation methods such as spline shape interpolation, a lofting surface body modeling and mesh generation method based on discrete surfaces is proposed. The method defines a simply connected domain on the lofted surface structure and determines the corner points to construct the corresponding topological quadrilateral, which is conformally mapped to the complex plane and the affine stretch transformation of the mapping parameter domain is calculated, and the conformal mapping of the simply connected domain is combined to obtain the Teichmuller space mapping and calculate the stretch quotient, and the target interpolation function is set to establish the relationship between the stretch quotient and the variables introduced in the simply connected domain, and the corresponding Beltrami coefficient is calculated, so as to obtain the design shape of the target interpolation simply connected domain, thereby finding a smooth mapping to achieve a continuous and uniform transition between the Jordan regions of two planar curved edges, and realizing fast and stable surface body lofting.

[0105] In addition, the second embodiment of the present invention also proposes a surface structure lofting device based on shape interpolation, including: a simply connected domain definition module, an affine stretching transformation module, a stretch quotient calculation module, an interpolation shape acquisition module and a lofting processing module. Among them, the simply connected domain definition module is used to define at least two simply connected domains on the lofted surface structure, determine the corner points of the simply connected domains, and construct corresponding topological quadrilaterals. The affine stretching transformation module is used to select any two of the simply connected domains, conformally map the topological quadrilaterals corresponding to each of the selected simply connected domains to a rectangular area on the complex plane, obtain the mapping parameter domain corresponding to each of the simply connected domains, and calculate the affine stretching transformation of the mapping parameter domain corresponding to the two simply connected domains. The stretch quotient calculation module is used to combine the conformal mappings corresponding to the two simply connected domains and the affine stretching transformation between the two simply connected domains to obtain the Teichmuller space mapping between the two simply connected domains, and calculate the stretch quotient between the two simply connected domains based on the Teichmuller space mapping. The interpolation shape obtaining module is used to determine the introduced variable between the two simply connected domains based on the expansion quotient, calculate the Beltrami coefficient between the two simply connected domains based on the expansion quotient and the introduced variable, and obtain the interpolation shape between the two simply connected domains based on the Beltrami coefficient. The lofting processing module is used to complete the lofting processing of the surface structure to be lofted according to the interpolation shape.

[0106] The surface structure lofting method based on shape interpolation implemented by the surface structure lofting device based on shape interpolation disclosed in the second embodiment of the present invention is as described in the first embodiment above, so it will not be described in detail here. Optionally, the various modules in the second embodiment and the above-mentioned other operations or functions are respectively for implementing the method described in the first embodiment, and the beneficial effects of the surface structure lofting device based on shape interpolation provided by this embodiment are the same as the beneficial effects of the surface structure lofting method based on shape interpolation provided by the first embodiment above, and for the sake of brevity, they are not repeated here.

[0107] The third embodiment of the present invention further proposes an electronic device, for example, comprising: at least one processing unit, and at least one storage unit, wherein the storage unit stores a computer program, and when the computer program is executed by the processing unit, the processing unit executes the method as described in the first embodiment, and the beneficial effects of the electronic device provided by this embodiment are the same as the beneficial effects of the surface structure lofting method based on shape interpolation provided by the first embodiment.

[0108] The fourth embodiment of the present invention also provides a computer-readable storage medium on which a computer program is stored. When the program is executed by a processor, the steps of the above method are implemented. The beneficial effects of the computer-readable storage medium provided by this embodiment are the same as the beneficial effects of the surface structure lofting method based on shape interpolation provided by the first embodiment.

[0109] Among them, computer-readable storage media may include, but are not limited to, any type of disk, including floppy disks, optical disks, DVDs, CD-ROMs, microdrives and magneto-optical disks, ROMs, RAMs, EPROMs, EEPROMs, DRAMs, VRAMs, flash memory devices, magnetic or optical cards, nanosystems (including molecular memory ICs), or any type of medium or device suitable for storing instructions and / or data.

[0110] It should be noted that, for the aforementioned method embodiments, for the sake of simplicity, they are all expressed as a series of action combinations, but those skilled in the art should be aware that the present application is not limited by the described order of actions, because according to the present application, certain steps can be performed in other orders or simultaneously. Secondly, those skilled in the art should also be aware that the embodiments described in the specification are all preferred embodiments, and the actions and modules involved are not necessarily required by the present application.

[0111] In the above embodiments, the description of each embodiment has its own emphasis. For parts that are not described in detail in a certain embodiment, reference can be made to the relevant descriptions of other embodiments.

[0112] In the several embodiments provided in the present application, it should be understood that the disclosed devices can be implemented in other ways. For example, the device embodiments described above are only schematic, such as the division of the units, which is only a logical function division. There may be other division methods in actual implementation, such as multiple units or components can be combined or integrated into another system, or some features can be ignored or not executed. Another point is that the mutual coupling or direct coupling or communication connection shown or discussed can be through some service interfaces, and the indirect coupling or communication connection of devices or units can be electrical or other forms.

[0113] The units described as separate components may or may not be physically separated, and the components shown as units may or may not be physical units, that is, they may be located in one place or distributed on multiple network units. Some or all of the units may be selected according to actual needs to achieve the purpose of the solution of this embodiment.

[0114] In addition, each functional unit in each embodiment of the present application may be integrated into one processing unit, or each unit may exist physically separately, or two or more units may be integrated into one unit. The above-mentioned integrated unit may be implemented in the form of hardware or in the form of software functional units.

[0115] If the integrated unit is implemented in the form of a software functional unit and sold or used as an independent product, it can be stored in a computer-readable memory. Based on this understanding, the technical solution of the present application is essentially or the part that contributes to the prior art or all or part of the technical solution can be embodied in the form of a software product, and the computer software product is stored in a memory, including a number of instructions to enable a computer device (which can be a personal computer, server or network device, etc.) to perform all or part of the steps of the method described in each embodiment of the present application. The aforementioned memory includes: U disk, read-only memory (ROM), random access memory (RAM), mobile hard disk, disk or optical disk and other media that can store program code.

[0116] A person skilled in the art may understand that all or part of the steps in the various methods of the above embodiments may be completed by entering a program to instruct the relevant hardware. The program may be stored in a computer-readable memory, and the memory may include: a flash drive, a read-only memory (ROM), a random access memory (RAM), a magnetic disk or an optical disk, etc.

[0117] The above is only an exemplary embodiment of the present disclosure, and the scope of the present disclosure cannot be limited thereto. That is, any equivalent changes and modifications made according to the teachings of the present disclosure are still within the scope of the present disclosure. After considering the specification and practicing the disclosure here, those skilled in the art will easily think of the implementation scheme of the present disclosure. This application is intended to cover any modification, use or adaptation of the present disclosure, which follows the general principles of the present disclosure and includes common knowledge or customary technical means in the technical field not recorded in the present disclosure. The description and examples are regarded as exemplary only, and the scope and spirit of the present disclosure are defined by the claims.

[0118] The technical features of the above embodiments may be combined arbitrarily. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0119] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is 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. A surface structure lofting method based on shape interpolation, characterized in that: include: Defining at least two simply connected domains on the surface structure to be lofted, and determining the corner points of each of the simply connected domains respectively, so as to construct a topological quadrilateral corresponding to each of the simply connected domains; Select any two of the simply connected domains, conformally map the topological quadrilateral corresponding to each of the selected simply connected domains into a rectangular area on the complex plane, obtain a mapping parameter domain corresponding to each of the simply connected domains, and calculate the affine stretching transformation of the mapping parameter domains corresponding to the two simply connected domains; Combining the conformal mappings corresponding to the two simply connected domains and the affine stretching transformation between the two simply connected domains to obtain a Teichmuller space mapping between the two simply connected domains, and calculating the stretch quotient between the two simply connected domains based on the Teichmuller space mapping; Determine an introduced variable between the two simply connected domains based on the stretch quotient, calculate a Beltrami coefficient between the two simply connected domains based on the stretch quotient and the introduced variable; and obtain an interpolation shape between the two simply connected domains based on the Beltrami coefficient; The lofting process of the surface structure to be lofted is completed according to the interpolation shape.

2. The surface structure lofting method based on shape interpolation according to claim 1 is characterized in that: The calculating the affine stretching transformation of the mapping parameter domain corresponding to the two simply connected domains includes: According to the conformal mapping of the topological quadrilateral to the rectangular region, a harmonic mapping is set to satisfy: Δh [m，n] =0; Among them, g m :S m →R m For the topological quadrilateral S m To the rectangular area R m The conformal mapping, g n :S n →R n For the topological quadrilateral S n To the rectangular area R n The conformal mapping, and are the boundaries of the two topological quadrilaterals, Δ is the Laplace operator defined on the complex plane; Determine the stretching affine transformation h of the mapping parameter domain corresponding to the two simply connected domains according to the harmonic mapping [m，n] .

3. The surface structure lofting method based on shape interpolation according to claim 2 is characterized in that: The step of combining the conformal mappings corresponding to the two simply connected domains and the affine stretching transformation between the two simply connected domains to obtain the Teichmuller space mapping between the two simply connected domains comprises: The stretching affine transformation h between the two simply connected domains [m，n] The conformal mappings g corresponding to the two simply connected domains respectively m and g n -1 Composite Mapping As described in Teichmuller space mapping.

4. The surface structure lofting method based on shape interpolation according to claim 1, characterized in that: The step of determining the introduced variable between the two simply connected domains based on the stretch quotient, and calculating the Beltrami coefficient between the two simply connected domains according to the stretch quotient and the introduced variable, comprises: The stretch quotient is made to change monotonically and linearly to determine the introduced variable between the two simply connected domains, an interpolation function is set to establish a relationship in which the stretch quotient changes with the change of the introduced variable, and the Beltrami coefficient corresponding to the stretch quotient is obtained according to the interpolation function.

5. The surface structure lofting method based on shape interpolation according to claim 1, characterized in that: The simply connected domain includes a plurality of triangular facets in a triangulated form, and the interpolation shape between two simply connected domains is obtained based on the Beltrami coefficients, including: The curved surface shape after mapping of each triangular facet is obtained according to the Beltrami coefficients, and all the mapped curved surface shapes are assembled and stitched to obtain the interpolation shape between the two simply connected domains.

6. The surface structure lofting method based on shape interpolation according to claim 5, characterized in that: The step of obtaining the surface shape of each triangular face after mapping according to the Beltrami coefficients includes: Acquire the vertex set and connection relationship of the triangular facets to determine an original triangulation mesh and a target triangulation mesh composed of the triangular facets on the two simply connected domains; Using a piecewise linear function to approximate the mapping of the original triangulated mesh to the target triangulated mesh, and obtaining a linear change of each triangular patch in the original triangulated mesh mapped to the target triangulated mesh; The area of ​​each triangular facet and the Beltrami coefficient are calculated according to the linear change to construct the surface shape after each triangular facet is mapped.

7. The surface structure lofting method based on shape interpolation according to claim 1, characterized in that: The step of completing the lofting process of the surface structure to be lofted according to the interpolation shape comprises: Arranging the mapping parameter domain corresponding to the simply connected domain according to the surface structure to be lofted, and normalizing it according to the direction of the distribution arrangement; Performing linear interpolation on the normalized mapping parameter domain according to the interpolation shape to form a parameter hexahedral region; The parameter hexahedral region is normalized in the direction of linear interpolation to obtain a unit cube mapping structure corresponding to the surface structure to be lofted, and a lofting configuration of the surface structure to be lofted is calculated according to the unit cube mapping structure.

8. A surface structure lofting device based on shape interpolation, characterized in that: include: A simply connected domain definition module, used to define at least two simply connected domains on the lofted surface structure, determine the corner points of the simply connected domains, and construct corresponding topological quadrilaterals; An affine stretching transformation module is used to select any two of the simply connected domains, conformally map the topological quadrilateral corresponding to each of the selected simply connected domains to a rectangular area on a complex plane, obtain a mapping parameter domain corresponding to each of the simply connected domains, and calculate the affine stretching transformation of the mapping parameter domains corresponding to the two simply connected domains; a stretch quotient calculation module, used for compounding the conformal mappings respectively corresponding to the two simply connected domains and the affine stretch transformation between the two simply connected domains to obtain a Teichmuller space mapping between the two simply connected domains, and calculating the stretch quotient between the two simply connected domains based on the Teichmuller space mapping; an interpolation shape obtaining module, used for determining an introduced variable between two simply connected domains based on the stretch quotient, calculating a Beltrami coefficient between the two simply connected domains according to the stretch quotient and the introduced variable; and obtaining an interpolation shape between the two simply connected domains based on the Beltrami coefficient; The lofting processing module is used to complete the lofting processing of the surface structure to be lofted according to the interpolation shape.

9. An electronic device, characterized in that: include: A memory and one or more processors connected to the memory, the memory storing a computer program, the processor being used to execute the computer program to implement the surface structure lofting method based on shape interpolation as described in any one of claims 1 to 7.

10. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores computer-executable commands, and the computer-executable commands are used to execute the surface structure lofting method based on shape interpolation as described in any one of claims 1 to 7.

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