Shape interpolation based lofting method for curved surface structure and application thereof

Abstract

This invention discloses a surface lofting method based on shape interpolation, comprising: defining at least two simply connected domains on the surface structure to be lofted; selecting two simply connected domains and conformally mapping them onto the complex plane to obtain corresponding mapping parameter domains; calculating the affine stretching transformation between the mapping parameter domains of the two simply connected domains; obtaining the Teichmüller space mapping by combining the conformal mapping and affine stretching transformation of the two simply connected domains and calculating the scaling quotient; calculating the corresponding Beltrami coefficients based on the scaling quotient, thereby obtaining the interpolation shape between the two simply connected domains; and completing the lofting process of the surface structure to be lofted based on the interpolation shape. It can achieve a continuous and uniform transition between two Jordan regions through interpolation, realizing fast and stable surface lofting.

Description

Technical Field

[0001] This application relates to the field of structural optimization design technology, and in particular to a method for lofting curved surface structures based on shape interpolation, a device for lofting curved surface structures based on shape interpolation, an electronic device, and a computer-readable storage medium. Background Technology

[0002] Lofting is the process of using multiple 2D objects as cutting planes to sweep along a predetermined direction to form complex 3D objects. It is commonly used in generating complex curved surfaces such as blades and is a frequently used operation for generating surfaces and surface bodies in mechanical structure design. Traditional lofting operations are mainly geared towards continuous parametric curves, such as NURBS curves, achieving curve masking calculations through order increase and node refinement. The core idea of ​​surface lofting is to interpolate each intermediate surface / curve using a set of compatible intermediate structures. Although the calculation principle for generating continuously parametric lofted surfaces is simple, it is often difficult to find smooth intermediate structures to achieve robust and convenient lofting functionality. Furthermore, interactively designing lofted surfaces is a tedious iterative process, dependent on the number, shape, and position of the lofted surfaces. Since the interpolation method is embedded in the computer, the designer has no control over it, thus consuming a significant amount of time to obtain the ideal lofted surface body configuration.

[0003] Therefore, how to achieve a continuous and uniform transition between 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] To overcome the shortcomings of the prior art, this invention provides a method and application for lofting curved surfaces based on shape interpolation, which can achieve a continuous and uniform transition between two Jordan regions through interpolation, thus realizing fast and stable lofting of curved surfaces.

[0005] On one hand, embodiments of the present invention propose a method for lofting curved surface structures based on shape interpolation, comprising: defining at least two simply connected domains on the curved surface structure to be lofted, and determining the corner points of each simply connected domain to construct a topological quadrilateral corresponding to each simply connected domain; selecting any two simply connected domains, conformally mapping the topological quadrilateral corresponding to each selected simply connected domain to a rectangular region on the complex plane to obtain the mapping parameter domain corresponding to each simply connected domain, 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 combined to obtain the Teichmüller space mapping between the two simply connected domains, and the scaling quotient between the two simply connected domains is calculated based on the Teichmüller space mapping; the introduced variable between the two simply connected domains is determined based on the scaling quotient and the introduced variable, and the Beltrami coefficient between the two simply connected domains is calculated based on the scaling quotient and the introduced variable; the interpolation shape between the two simply connected domains is obtained based on the Beltrami coefficient; and the lofting process of the surface structure to be lofted is completed based on the interpolation shape.

[0006] In one embodiment of the present invention, calculating the affine stretching transformation of the mapping parameter domains corresponding to the two simply connected domains includes: setting a harmonic mapping that satisfies the conformal mapping from the topological quadrilateral to the rectangular region: Among them, g m :S m →R m For the topological quadrilateral S m To the rectangular region R m The conformal mapping, g n :S n →R n For the topological quadrilateral S n To the rectangular region R n The conformal mapping, and Let Δ be the boundary of the two topological quadrilaterals, and let Δ be the Laplacian 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] .

[0007] In one embodiment of the present invention, 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 Teschmüller space mapping between the two simply connected domains includes: combining the stretching affine transformation h between the two simply connected domains... [m，n] The conformal mapping g corresponding to the two simply connected domains respectively mand g n -1 Composite mapping obtained by phase composite As the Teschmüller space mapping.

[0008] In one embodiment of the present invention, determining the introduced variable between two simply connected domains based on the scaling quotient, and calculating the Beltrami coefficient between the two simply connected domains based on the scaling quotient and the introduced variable, includes: making the scaling quotient monotonically linear to determine the introduced variable between the two simply connected domains, setting an interpolation function to establish the relationship between the scaling quotient and the introduced variable, and obtaining the Beltrami coefficient corresponding to the scaling quotient based on the interpolation function.

[0009] In one embodiment of the present invention, the simply connected domain includes a plurality of triangular facets in the form of triangulation, and obtaining the interpolation shape between two simply connected domains based on the Beltrami coefficient includes: obtaining the surface shape mapped by each of the triangular facets according to the Beltrami coefficient, 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, obtaining the surface shape mapped to each of the triangular facets based on the Beltrami coefficient includes: acquiring the vertex set and connection relationship of the triangular facets to determine the original triangulation mesh and the target triangulation mesh composed of the triangular facets on two simply connected domains; approximating the mapping from the original triangulation mesh to the target triangulation mesh using a piecewise linear function to obtain the linear transformation of each triangular facet in the original triangulation mesh to the target triangulation mesh; and calculating the area of ​​each triangular facet and the Beltrami coefficient based on the linear transformation to construct the surface shape mapped to each of the triangular facets.

[0011] In one embodiment of the present invention, the lofting process 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 distribution of 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 parametric hexahedral region; normalizing the parametric hexahedral region according to the direction of the linear interpolation to obtain the 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, this invention also proposes a surface structure lofting device based on shape interpolation, comprising: a single connected domain definition module, used to define at least two single connected domains on the lofted surface structure, determine the corner points of the single connected domains, and construct corresponding topological quadrilaterals; an affine stretching transformation module, used to select any two single connected domains, conformally map the topological quadrilateral corresponding to each selected single connected domain to a rectangular region on the complex plane, obtain the mapping parameter domain corresponding to each single connected domain, and calculate the affine stretching transformation of the mapping parameter domains corresponding to the two single connected domains; and a stretching quotient calculation module, used to calculate the stretching quotient of the two single connected domains respectively. The corresponding conformal mapping and the affine stretching transformation between the two simply connected domains are combined to obtain the Teichmüller space mapping between the two simply connected domains, and the scaling quotient between the two simply connected domains is calculated based on the Teichmüller space mapping; the interpolation shape acquisition module is used to determine the introduced variables between the two simply connected domains based on the scaling quotient, and calculate the Beltrami coefficients between the two simply connected domains based on the scaling quotient and the introduced variables; the interpolation shape between the two simply connected domains is obtained based on the Beltrami coefficients; the lofting processing module is used to complete the lofting processing of the surface structure to be lofted according to the interpolation shape.

[0013] In another aspect, embodiments of the present invention also propose an electronic device, comprising: a memory and one or more processors connected to the memory, the memory storing a computer program, and the processors being configured to execute the computer program to implement the shape interpolation-based surface structure lofting method as described in any of the above embodiments.

[0014] In another aspect, embodiments of the present invention also provide a computer-readable storage medium storing computer-executable instructions for performing the shape interpolation-based surface structure lofting method as described in any of the above embodiments.

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

[0016] By combining mathematical concepts such as conformal geometry and Teschmüller space mapping with computational methods such as spline shape interpolation, a method for lofting surface modeling and mesh generation based on discrete surfaces is proposed. This method defines a single connected domain on the lofted surface structure and determines the corner points to construct the corresponding topological quadrilaterals. After conformal mapping to the complex plane, the affine stretching transformation of the mapping parameter domain is calculated. This transformation is then combined with the conformal mapping of the single connected domain to obtain the Teschmüller space mapping and calculate the scaling quotient. A target interpolation function is set to establish the relationship between the scaling quotient and the variables introduced in the single connected domain. The corresponding Beltrami coefficients are calculated, thereby obtaining the design shape of the target interpolated single connected domain. This method finds a smooth mapping to achieve a continuous and uniform transition between two planar curved Jordan regions, realizing fast and stable surface lofting. Attached Figure Description

[0017] The accompanying drawings, which are included to provide a further understanding of the invention and form part of this invention, illustrate exemplary embodiments of the invention and are used to explain the invention, but do not constitute an undue limitation of the invention. In the drawings:

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

[0019] Figure 2 This is a schematic diagram illustrating the mapping principle between two adjacent lofted surfaces provided in an embodiment of the present invention;

[0020] Figure 3 A schematic diagram illustrating the process of constructing the surface shape of a triangular patch mapping based on the Beltrami coefficient, as provided in an embodiment of the present invention;

[0021] Figure 4 This is a schematic diagram of lofted surface modeling and parameterization provided in an embodiment of the present invention. Detailed Implementation

[0022] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other. The present invention will now be described with reference to the accompanying drawings and embodiments.

[0023] To enable those skilled in the art to better understand the technical solutions of the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments, and should all fall within the protection scope of the present invention.

[0024] It should be noted that the terms "first," "second," etc., used in the specification, claims, and accompanying drawings of this invention are applicable in distinguishing similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such terms can be used interchangeably where appropriate so that the embodiments of the invention described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover a non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or applicable to such processes, methods, products, or apparatus.

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

[0026] like Figure 1 As shown, the first embodiment of the present invention proposes a method for lofting curved surface structures based on shape interpolation, including, for example, the following steps: Step S1, defining at least two simply connected domains on the curved surface structure to be lofted, and determining the corner points of each simply connected domain to construct a topological quadrilateral corresponding to each simply connected domain; Step S2, selecting any two simply connected domains, and conformally mapping the topological quadrilateral corresponding to each selected simply connected domain to a rectangular region on the complex plane to obtain the mapping parameter domain corresponding to each simply connected domain, and calculating the affine stretching transformation of the mapping parameter domain corresponding to the two simply connected domains; Step S3, transferring the two simply connected domains... The conformal mapping corresponding to each of the connected domains and the affine stretching transformation between the two simply connected domains are combined to obtain the Teichmüller space mapping between the two simply connected domains, and the scaling quotient between the two simply connected domains is calculated based on the Teichmüller space mapping; Step S4, the introduced variable between the two simply connected domains is determined based on the scaling quotient, and the Beltrami coefficient between the two simply connected domains is calculated based on the scaling quotient and the introduced variable; the interpolation shape between the two simply connected domains is obtained based on the Beltrami coefficient; Step S5, the lofting process of the surface structure to be lofted is completed according to the interpolation shape.

[0027] Specifically, in combination Figure 2 As shown, for example, two planar curved edges Jordan regions S are defined on the surface structure to be lofted. m and S n , will S m and S n By selecting corner points to transform them into topological quadrilaterals, the two topological quadrilaterals are conformally mapped onto the parameter domain R. m and Rn Inside, the ratio of the rectangle's x and y coordinates, a m and a n That is, S m and S n The conformal mode. Further utilizing a m and a n Calculate from S m and S n The scaling quotient is obtained, and the stretching affine transformation h between the two parameter domains is obtained. [m，n] A linear function of the scaling quotient with respect to parameter t is established, and the Beltrami coefficient is calculated using the scaling quotient value corresponding to each parameter t. Then, the interpolation shape is determined based on the corresponding Beltrami coefficient. When parameter t is within the 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 represented by triangulation as follows: The interpolation parameter t takes values ​​in the range [0, 1]. For two Jordan regions S m and S n Smooth shape interpolation between these elements must satisfy the following conditions:

[0029] (1) When t∈[0,1], we have f t＝0 =Id and f t＝1 =f. For the interpolation of the mapping, it must satisfy that when t=0, the mapping is to transform 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 mapping, i.e., f t＝1 =f:S m →S n .

[0030] (2) Pointwise bounded nonconformal distortion. Let point p be S. m A point in the equation, i.e., p∈S m K(p) is the non-conformal distortion at point p, i.e., the scaling quotient. And when the mapping at time t is interpolated, it is... When, the corresponding ones are If it is the expansion quotient at this time, then we have It should be noted that if in f [m，n] :S m →S n There is a special case where K(f) ≡ 1 for any point, meaning that the mapping f is a conformal mapping. Therefore, for... have Right now At this point, it is also necessary to maintain a conformal mapping.

[0031] (3) Symmetry. Let f -1 :S m →S n This is the inverse process of mapping f, and From S n To S m If the interpolation result at time t is obtained using the same method, then we have: In other words, regardless of whether it is from S m To S n Or from S n To S m The interpolation results obtained for t are also one-to-one correspondences, and have

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

[0033] Therefore, in step S1, for example, in the two Jordan regions S respectively m and S n Find the corresponding counterclockwise distributed corner points on the top, that is... and Construct S m and S n Based on 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 respectively mapped to R through conformal mapping. m and R n Within two planar rectangles, two rectangles R are obtained 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] To obtain continuous shape interpolation, a harmonic mapping must be found that satisfies:

[0038]

[0039] in, and Let Δ be the boundary between two topological quadrilaterals, and let Δ be the Laplacian operator defined on the complex plane. The stretching affine transformation h... [m，n] In the rectangle, the ordinate of each point remains unchanged, while the abscissa changes proportionally, that is:

[0040]

[0041] Therefore, as Figure 2 As shown, for composite mapping Due to g m and g n -1 It is a conformal mapping, and h [m，n] It is harmonized, therefore f [m，n] It is also harmonized. Furthermore, due to g m and g n -1 If it is a conformal mapping, then the mapping f [m，n] Conformal distortion and h [m，n] The conformal distortion is the same. At the same time, it can be calculated that... The Beltrami coefficient is:

[0042]

[0043] This means:

[0044]

[0045] Similarly, we can obtain:

[0046]

[0047] Due to h [m，n] :R m →R n It is a stretching affine transformation with a stretch quotient of K. z (f)=κ mn That is, a n / a m (a n >a m ) or a m / a n (a n ≤a m Therefore, f [m，n] The Beltrami coefficient can be calculated as:

[0048]

[0049] At this point, the Teichmüller distance of the mapping, which is the maximum quasi-conformal dilatation between the two parameter domains (and further developed to the extreme value mapping within the two planar 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 and one stretched affine transformation (Teichmüller mapping) g m g n -1 and h [m，n] Phase recombination yields from S m and S n Teichmüller mapping:

[0052]

[0053] The mapping f can be calculated [m，n] After obtaining the scaling quotient K(f), a control variable t∈[0,1] is introduced into the two Jordan region interpolation problems, and a suitable interpolation function is selected to establish the relationship K between the scaling quotient and the variable t. t (f) After that, the relationship μ between the Beltrami coefficient and the variable t is calculated. t (f).

[0054] For two Jordan regions S on the complex planem and S n ,according to Figure 2 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 retractable supplier is

[0058]

[0059] In step S4, when t∈[0,1], if the scaling quotient changes monotonically with t, we can first assume that this change is linear, i.e., K t (f)=(K(f)-1)·t+1, and simultaneously according to K t (f) We can obtain the Beltrami coefficient of the mapping at this time as follows:

[0060]

[0061] Define a normalization coefficient Make it satisfy:

[0062]

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

[0064]

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

[0066] Furthermore, for example, input two Jordan regions S in the form of triangulation.m and S n S m To S n mapping relationship For ease of calculation and representation, for S m and S n The triangulation representation needs to unify the index order in its connection relationships, that is, for S m ={X m F m} and S n ={X n F n}, there is F m =F n Therefore, S will be used subsequently. m and S n The connection relationships between nodes are all uniformly represented by F. If the triangulation of the input surface does not correspond, the mapping relationship f between the two surfaces can be used. [m，n] S m Mapping coordinates on S n Above, and continue to use S m On the triangulation index F m .

[0067] The above steps describe a method for calculating the scaling quotient and Beltrami coefficient at any intermediate time t using the mapping relationship between two topological quadrilaterals. Theoretically, given the Beltrami coefficient μ... t After (f), there must exist a smooth mapping f. t Correspondingly, in actual computation, the desired piecewise linear mapping cannot be obtained by directly solving the Beltrami equation. Furthermore, since the Beltrami coefficients in the actual calculation of piecewise mapping depend on the triangular facets (i.e., the connectivity F) rather than the coordinate points, the resulting equations cannot be solved for discrete triangulated meshes where the number of points is often only half the number of faces. Therefore, for example, the solution to the mapping is divided into two steps: first, calculating the shape of each triangular facet after mapping under a specified scaling quotient and Beltrami coefficients; then, assembling and stitching together all the triangular facets to obtain the target interpolation result.

[0068] Furthermore, using S m ={V m Let F represent region S m On the triangular mesh, where X m It is the set of vertices of triangles, and F represents the connection relationship between the triangles. f = u + iv is from the original mesh S. m ={V m ,F}to the target meshS n ={V nThe mapping of F}, where the grid S m and S n The number of vertices is the same, and they have the same connection relationship F. A piecewise linear function is used to approximate the mapping f, specifically, f 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 It is the counterclockwise sorted vertex of triangle F.

[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 the mapping is v = g + ih, and the point after the mapping is w = s + it, then:

[0072]

[0073] Then, for each triangle, we can obtain the following based on linear transformation:

[0074]

[0075]

[0076] Among them are:

[0077]

[0078]

[0079] Among them, A F This is the area of ​​the triangular facet. Simultaneously, the Beltrami coefficient can be calculated on this triangle as:

[0080]

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

[0082]

[0083] Solve for v′ kx and v′ kx .

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

[0085] Specifically, in the two-dimensional complex plane, the rotation and scaling matrices of a vector can be written in the following form:

[0086]

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

[0088]

[0089] Where E represents the boundary relationship of the triangle, E = {{i, j}, {j, k}, {k, i}}. Substituting the matrix form from the above into the expression, we can obtain the optimized formulation of each term in the coordinate transformation matrix as follows:

[0090]

[0091] in,

[0092]

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

[0094]

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

[0096]

[0097]

[0098] The formula is about variables The quadratic positive definite optimization problem can be written in classic matrix form, i.e.

[0099]

[0100] in, The coefficient matrix is ​​assembled, with each row corresponding to a summation term in the column; variables Let N be a 2N×1 vector, where N is the number of coordinate points in the triangulation:

[0101]

[0102] b is a (M+2C)×1 column vector, where M is the number of terms summed in the first half of the equation, and C is the number of fixed points. From the matrix form, it can be seen that this optimization problem can be solved using the conjugate gradient method given some vertices. Thus, we have obtained the Teichmüller mapping between the Jordan regions on the two complex planes and the scaling quotient of each corresponding triangular facet, and calculated the corresponding Beltrami coefficients by interpolating the scaling quotients. Further, based on the Beltrami coefficients interpolated from the intermediate states, we solve for the shape of each triangular facet, and then perform ARAP stitching on all triangles. By solving a quadratic positive definite optimization problem, we obtain the shape of the intermediate transition surface and the corresponding triangulation.

[0103] Furthermore, lofted surface modeling and parametric methods, such as Figure 4 As shown, the lofted surfaces are conformally mapped onto the rectangular parameter domain, and all lofted surfaces are arranged according to the distribution of the normalized T-number. Next, the U-direction interpolation is performed using a linear interpolation method to form a parametric hexahedral region. Then, the continuous parameters in the U-direction are normalized to obtain the mapping from the lofted surface to the unit cube. Finally, the configuration of the lofted surface is calculated based on this mapping.

[0104] In summary, the first embodiment of this invention proposes a method for lofting surface structures based on shape interpolation. By combining mathematical concepts such as conformal geometry and Teichmüller space mapping with calculation methods such as spline shape interpolation, a method for modeling and meshing lofted surface bodies based on discrete surfaces is proposed. This method defines a single connected domain on the lofted surface structure and determines the corner points to construct the corresponding topological quadrilaterals. After conformal mapping to the complex plane, the affine stretching transformation of the mapping parameter domain is calculated. This transformation is then combined with the conformal mapping of the single connected domain to obtain the Teichmüller space mapping and calculate the scaling quotient. A target interpolation function is set to establish the relationship between the scaling quotient and the variables introduced in the single connected domain. The corresponding Beltrami coefficients are calculated, thereby obtaining the design shape of the target interpolated single connected domain. This method finds a smooth mapping to achieve a continuous and uniform transition between two planar curved Jordan regions, realizing fast and stable surface body lofting.

[0105] Furthermore, the second embodiment of the present invention proposes a surface structure lofting device based on shape interpolation, comprising: a single connected domain definition module, an affine stretching transformation module, a scaling quotient calculation module, an interpolation shape acquisition module, and a lofting processing module. The single connected domain definition module defines at least two single connected domains on the lofted surface structure, determines the corner points of the single connected domains, and constructs corresponding topological quadrilaterals. The affine stretching transformation module selects any two single connected domains, conformally maps the topological quadrilateral corresponding to each selected single connected domain to a rectangular region on the complex plane, obtains the mapping parameter domain corresponding to each single connected domain, and calculates the affine stretching transformation of the mapping parameter domains corresponding to the two single connected domains. The scaling quotient calculation module combines the conformal mappings corresponding to the two single connected domains and the affine stretching transformation between the two single connected domains to obtain the Teichmüller space mapping between the two single connected domains, and calculates the scaling quotient between the two single connected domains based on the Teichmüller space mapping. The interpolation shape acquisition module is used to determine the introduced variable between the two simply connected domains based on the scaling quotient, calculate the Beltrami coefficient between the two simply connected domains based on the scaling 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 based on the interpolation shape.

[0106] The shape interpolation-based surface structure lofting device disclosed in the second embodiment of the present invention implements the shape interpolation-based surface structure lofting method as described in the first embodiment above, and therefore will not be described in detail here. Optionally, each module and the other operations or functions described above in the second embodiment are for implementing the method described in the first embodiment, and the beneficial effects of the shape interpolation-based surface structure lofting device provided in this embodiment are the same as the beneficial effects of the shape interpolation-based surface structure lofting method provided in the first embodiment above. For the sake of brevity, they will not be repeated here.

[0107] The third embodiment of the present invention also proposes an electronic device, for example including: 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 performs the method described in the first embodiment, and the beneficial effects of the electronic device provided in this embodiment are the same as the beneficial effects of the surface structure lofting method based on shape interpolation provided in the first embodiment.

[0108] The fourth embodiment of the present invention also provides a computer-readable storage medium storing a computer program thereon, which, when executed by a processor, implements the steps of the above-described method. The beneficial effects of the computer-readable storage medium provided in this embodiment are the same as those of the shape interpolation-based surface structure lofting method provided in the first embodiment.

[0109] The computer-readable storage medium may include, but is not limited to, any type of disk, including floppy disks, optical disks, DVDs, CD-ROMs, microdrives, as well as magneto-optical disks, ROMs, RAMs, EPROMs, EEPROMs, DRAMs, VRAMs, flash memory devices, magnetic cards 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 sake of simplicity, the foregoing method embodiments are all described as a series of actions. However, those skilled in the art should understand that this application is not limited to the described order of actions, as some steps may be performed in other orders or simultaneously according to this application. Furthermore, those skilled in the art should also understand that the embodiments described in the specification are preferred embodiments, and the actions and modules involved are not necessarily essential to this application.

[0111] In the above embodiments, the descriptions of each embodiment have different focuses. For parts not described in detail in a certain embodiment, please refer to the relevant descriptions in other embodiments.

[0112] In the several embodiments provided in this application, it should be understood that the disclosed apparatus can be implemented in other ways. For example, the apparatus embodiments described above are merely illustrative; for instance, the division of units is only a logical functional division, and in actual implementation, there may be other division methods. For example, multiple units or components may be combined or integrated into another system, or some features may be ignored or not executed. Furthermore, the coupling or direct coupling or communication connection shown or discussed may be through some service interface; the indirect coupling or communication connection between devices or units may be electrical or other forms.

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

[0114] Furthermore, the functional units in the various embodiments of this application can be integrated into one processing unit, or each unit can exist physically separately, or two or more units can be integrated into one unit. The integrated unit can be implemented in hardware or as a software functional unit.

[0115] If the integrated unit is implemented as a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage device (CMD). Based on this understanding, the technical solution of this application, in essence, 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. This computer software product is stored in a memory and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of this application. The aforementioned memory includes various media capable of storing program code, such as USB flash drives, read-only memory (ROM), random access memory (RAM), portable hard drives, magnetic disks, or optical disks.

[0116] Those skilled in the art will understand that all or part of the steps in the various methods of the above embodiments can be implemented by a program instructing related hardware. The program can be stored in a computer-readable storage medium, which 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 foregoing description is merely an exemplary embodiment of this disclosure and should not be construed as limiting the scope of this disclosure. Any equivalent changes and modifications made in accordance with the teachings of this disclosure shall still fall within the scope of this disclosure. Those skilled in the art will readily conceive of embodiments of this disclosure upon considering the specification and practicing the disclosure herein. This application is intended to cover any variations, uses, or adaptations of this disclosure that follow the general principles of this disclosure and include common knowledge or customary techniques in the art not described herein. The specification and embodiments are to be considered exemplary only, and the scope and spirit of this disclosure are defined by the claims.

[0118] The technical features of the above embodiments can be combined in any way. For the sake of brevity, 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] Those skilled in the art will readily understand that the above description is merely 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 within the scope of protection of the present invention.

Claims

1. A method for lofting curved surface structures based on shape interpolation, characterized in that, in, The curved surface structure is a blade, and the method includes: Define at least two simply connected domains on the surface structure to be lofted, and determine the corner points of each simply connected domain to construct a topological quadrilateral corresponding to each simply connected domain; Select any two simply connected domains, and conformally map the topological quadrilateral corresponding to each selected simply connected domain to a rectangular region on the complex plane to obtain the mapping parameter domain corresponding to each simply connected domain, and calculate the affine stretching transformation of the mapping parameter domain corresponding to the two simply connected domains. 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 Teschmüller space mapping between the two simply connected domains, and the stretching quotient between the two simply connected domains is calculated based on the Teschmüller space mapping. Based on the scaling quotient, the introduced variable between the two simply connected domains is determined, and the Beltrami coefficient between the two simply connected domains is calculated according to the scaling quotient and the introduced variable; the interpolation shape between the two simply connected domains is obtained based on the Beltrami coefficient. The lofting process of the surface structure to be lofted is completed based on the interpolation shape.

2. The method for lofting curved surfaces based on shape interpolation according to claim 1, characterized in that, The calculation of the affine stretching transformation of the mapping parameter domain corresponding to the two simply connected domains includes: The harmonic mapping is set according to the conformal mapping from the topological quadrilateral to the rectangular region, satisfying: ; in, The topological quadrilateral to the rectangular area The conformal mapping, The topological quadrilateral to the rectangular area The conformal mapping, and For the boundaries of the two aforementioned topological quadrilaterals, Let Laplace be the operator defined on the complex plane; The affine stretching transformation of the mapping parameter domain corresponding to the two simply connected components is determined based on the harmonic mapping. .

3. The method for lofting curved surface structures based on shape interpolation according to claim 2, 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 Teichmüller space mapping between the two simply connected domains includes: The affine stretching transformation between the two simply connected components The conformal mappings corresponding to the two simply connected domains respectively and Composite mapping obtained by phase composite As the Teschmüller space mapping.

4. The method for lofting curved surface structures based on shape interpolation according to claim 1, characterized in that, The step of determining the introduced variable between the two simply connected components based on the scaling quotient, and calculating the Beltrami coefficient between the two simply connected components based on the scaling quotient and the introduced variable, includes: The scaling quotient is monotonically linearly varied to determine the introduced variable between the two simply connected domains. An interpolation function is set to establish the relationship between the scaling quotient and the introduced variable. The Beltrami coefficient corresponding to the scaling quotient is obtained according to the interpolation function.

5. The method for lofting curved surface structures based on shape interpolation according to claim 1, characterized in that, The simply connected domain comprises multiple triangular facets in the form of triangulation, and obtaining the interpolation shape between two simply connected domains based on the Beltrami coefficient includes: The surface shape after mapping each of the triangular facets is obtained based on the Beltrami coefficients. All the mapped surface shapes are assembled and stitched together to obtain the interpolated shape between the two simply connected domains.

6. The method for lofting curved surface structures based on shape interpolation according to claim 5, characterized in that, The step of obtaining the surface shape after mapping each triangular facet based on the Beltrami coefficient includes: Obtain the vertex set and connection relationship of the triangular facets to determine the original triangulation mesh and the target triangulation mesh composed of the triangular facets on the two simply connected domains; The mapping from the original triangulation mesh to the target triangulation mesh is approximated by a piecewise linear function, resulting in a linear transformation of each triangular facet in the original triangulation mesh onto the target triangulation mesh. The area of ​​each triangular facet and the Beltrami coefficient are calculated based on the linear transformation to construct the surface shape mapped from each triangular facet.

7. The method for lofting curved surface structures based on shape interpolation according to claim 1, characterized in that, The lofting process for the surface structure to be lofted based on the interpolated shape includes: The mapping parameter domains corresponding to the simply connected domains are arranged according to the distribution of the surface structure to be lofted, and then normalized according to the direction of the distribution arrangement. Linear interpolation is performed on the normalized mapping parameter domain according to the interpolation shape to form a parametric hexahedral region; The parameter hexahedral region is normalized according to the direction of linear interpolation to obtain the unit cube mapping structure corresponding to the surface structure to be lofted, and the lofting configuration of the surface structure to be lofted is calculated based on the unit cube mapping structure.

8. A surface structure lofting device based on shape interpolation, characterized in that, in, The curved structure is a blade, including: A simply connected domain definition module is used to define at least two simply connected domains on the surface structure to be lofted, determine the corner points of the simply connected domains, and construct the corresponding topological quadrilaterals. The affine stretching transformation module is used to select any two simply connected domains, conformally map the topological quadrilateral corresponding to each selected simply connected domain to a rectangular region on the complex plane, obtain the mapping parameter domain corresponding to each simply connected domain, and calculate the affine stretching transformation of the mapping parameter domain corresponding to the two simply connected domains. The scaling 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 Teichmüller space mapping between the two simply connected domains, and to calculate the scaling quotient between the two simply connected domains based on the Teichmüller space mapping. An interpolation shape obtaining module is used to determine the introduced variable between two simply connected domains based on the scaling quotient, calculate the Beltrami coefficient between the two simply connected domains based on the scaling 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.

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 processors executing the computer program to implement the shape interpolation-based surface structure lofting method as described in any one of claims 1-7.

10. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores computer-executable commands for performing the shape interpolation-based surface structure lofting method as described in any one of claims 1-7.

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