Complex tubular structure modeling method based on 2D segmented algebraic spline
By using a 2D piecewise algebraic spline-based method, leveraging the Frenet framework and implicit field generation technology, the problem of numerous control points and low efficiency in existing technologies is solved, enabling efficient modeling and high-quality visualization of complex tubular structures.
Patent Information
- Application Number
- CN202510983633.7
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-07-17
- Publication Date
- 2025-10-31
AI Technical Summary
Existing tubular structure modeling techniques suffer from problems such as a large number of control points, low model generation efficiency, and insufficient flexibility when dealing with highly detailed or complex geometric models.
A method based on 2D piecewise algebraic splines is adopted. By selecting control points in a two-dimensional plane to form cross-sectional control polygons, zero-order and higher-order basis functions are constructed. Euclidean transformation and implicit field generation are performed in combination with the Frenet framework. The MarchingCubes algorithm is used to extract the triangular mesh model for visualization rendering and editing.
It significantly reduces the number of control points, improves modeling accuracy and efficiency, and enables high-quality modeling and visualization of complex tubular structures. It is suitable for complex tubular objects such as blood vessels, pipelines, and flexible structures.
Smart Images
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Abstract
Description
Technical Field
[0001] This invention relates to the fields of geometric modeling and computer graphics, and in particular to a method for modeling complex tubular structures based on 2D piecewise algebraic splines. Background Technology
[0002] Existing tubular structure modeling techniques mainly include polygon-based modeling, parametric modeling, and tensor product modeling methods, but all of them have significant limitations.
[0003] Polygonal modeling methods use vertices, edges, and faces to construct complex 3D shapes. When dealing with highly detailed or complex geometric models, the fineness of the mesh affects the model. Parametric modeling methods emphasize controlling the geometry through parameters. Although this method has a certain degree of flexibility, it requires frequent modification of variable parameters. While tensor product modeling methods can generate complex geometric models, they require a large number of control points when generating complex models, and the selection of control points for complex models is a significant workload. Summary of the Invention
[0004] To overcome the shortcomings of existing technologies, the purpose of this invention is to provide a modeling method for complex tubular structures based on 2D piecewise algebraic splines, which greatly reduces the number of control points and provides an efficient solution for the batch generation of complex industrial pipes.
[0005] To achieve the above objectives, the present invention provides the following solution:
[0006] A method for modeling complex tubular structures based on 2D piecewise algebraic splines, comprising:
[0007] Select several control points in a two-dimensional plane and connect them in a counterclockwise order to form a cross-sectional control polygon;
[0008] Based on the cross-sectional control polygon, a zero-order 2D piecewise algebraic spline basis function is constructed, and a higher-order basis function is constructed through integral recursion; wherein the smoothing parameter of the higher-order basis function is used to adjust the smoothness of the cross-sectional edge.
[0009] A guide curve is defined in three-dimensional space, and the Frenet frame at each parameter point of the guide curve is calculated.
[0010] The cross-section control polygons are mapped to the corresponding Frenet frame at each parameter point by Euclidean transformation to obtain multi-section control polygons distributed along the guide curve.
[0011] Using the multi-section control polygon and the corresponding 2D piecewise algebraic spline basis function as parameters, an implicit field is constructed, and an implicit representation of the complex tubular structure is generated.
[0012] The MarchingCubes algorithm is applied to the implicit field to extract isosurfaces, thereby obtaining an explicit triangular mesh model of the complex tubular structure.
[0013] The triangular mesh model is then visualized, rendered, or edited.
[0014] Preferably, the zeroth-order 2D piecewise algebraic spline basis function The expression is:
[0015]
[0016] Where, χ Δ (x,y) is the characteristic function, indicating that the point (x,y) controls the polygon Δ. x is the x-coordinate of a point, and y is the y-coordinate of a point.
[0017] Preferably, the expression for the higher-order basis function is:
[0018]
[0019] Where δ is a smoothing parameter used to control the smoothness of the tubular cross-section, s is the abscissa of a point on the two-dimensional plane, t is the ordinate of a point on the two-dimensional plane, and n is the order of the spline, representing the degree of function smoothing.
[0020] Preferably, the Euclidean transformation process is as follows: first, rotate around the tangent direction of the Frenet frame, and then translate along the position vector of the guide curve.
[0021] Preferably, the connection gap distance D between adjacent cross-section control polygons is 0.
[0022] Preferably, two adjacent cross-section control polygons share at least one identical control point on their common edge to ensure multi-section connectivity.
[0023] Preferably, after performing the MarchingCubes algorithm on the implicit field to extract isosurfaces and obtain an explicit triangular mesh model of the complex tubular structure, the method further includes:
[0024] The VTK library is invoked to perform smoothing, denoising, and texture mapping processing on the triangular mesh model.
[0025] According to specific embodiments provided by the present invention, the present invention discloses the following technical effects:
[0026] This invention can flexibly control the cross-sectional shape and structural smoothness, and combined with the Frenet framework, it can achieve precise arrangement and seamless connection of cross-sections along spatial curves. By constructing implicit field generation of three-dimensional models and extracting high-quality meshes, it improves the accuracy, efficiency and geometric continuity of modeling, and is suitable for high-quality modeling and visualization of complex tubular objects such as blood vessels, pipelines and flexible structures. Attached Figure Description
[0027] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.
[0028] Figure 1 A flowchart of the method provided in an embodiment of the present invention;
[0029] Figure 2 This is a schematic diagram of a first complex pipe model provided for an embodiment of the present invention;
[0030] Figure 3 This is a second schematic diagram of a complex pipe model provided in an embodiment of the present invention;
[0031] Figure 4 This is a schematic diagram of the third complex pipe model provided in an embodiment of the present invention;
[0032] Figure 5 This is a fourth schematic diagram of a complex pipe model provided in an embodiment of the present invention;
[0033] Figure 6 This is a fifth schematic diagram of a complex pipe model provided in an embodiment of the present invention;
[0034] Figure 7 The sixth schematic diagram of the complex pipe model provided in this embodiment of the invention;
[0035] Figure 8 The seventh schematic diagram of a complex pipe model provided for an embodiment of the present invention. Detailed Implementation
[0036] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.
[0037] The purpose of this invention is to provide a modeling method for complex tubular structures based on 2D piecewise algebraic splines, which greatly reduces the number of control points and provides an efficient solution for the batch generation of complex industrial pipes.
[0038] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.
[0039] Figure 1 The method flowchart provided in the embodiments of the present invention is as follows: Figure 1 As shown, this invention provides a method for modeling complex tubular structures based on 2D piecewise algebraic splines, including:
[0040] Step 100: Select several control points in the two-dimensional plane and connect them in a counterclockwise order to form a cross-sectional control polygon;
[0041] Step 200: Construct zero-order 2D piecewise algebraic spline basis functions based on the cross-section control polygon, and construct higher-order basis functions through integral recursion; wherein the smoothing parameter of the higher-order basis functions is used to adjust the smoothness of the cross-section edges;
[0042] Step 300: Determine a guide curve in three-dimensional space and calculate the Frenet frame at each parameter point of the guide curve;
[0043] Step 400: Map the cross-section control polygons to the corresponding Frenet frame at each parameter point using Euclidean transformation to obtain multiple cross-section control polygons distributed along the guide curve;
[0044] Step 500: Using the multi-section control polygon and the corresponding 2D piecewise algebraic spline basis function as parameters, construct an implicit field and generate an implicit representation of the complex tubular structure;
[0045] Step 600: Perform the MarchingCubes algorithm on the implicit field to extract isosurfaces and obtain an explicit triangular mesh model of the complex tubular structure;
[0046] Step 700: Visualize and render the triangular mesh model or perform subsequent editing.
[0047] Specifically, the steps for using 2D piecewise algebraic splines in complex tube modeling are as follows:
[0048] 1. Based on the target tubular structure, select cross-sectional control points and construct control polygons.
[0049] Determine the cross-sectional shape: Based on the target tubular structure, select an appropriate number of control points in a two-dimensional plane. These control points are arranged counterclockwise to form a control polygon.
[0050] Foundation for basis functions: For the initial control polygon, the zeroth-order basis function is defined as follows: Where, χ Δ (x,y) is the characteristic function, indicating that the point (x,y) controls the polygon Δ.
[0051] 2. Adjust the smoothing parameter δ to control the smoothness of the tubular cross-section.
[0052] Higher-order basis functions are defined recursively through integration to achieve smoothing of the cross-sectional edges: (n>0). Here, δ is the parameter controlling polygon smoothing, and this function has two key properties: It is a piecewise polynomial function, which can flexibly describe complex shapes. It is C n-1 The continuity ensures the continuity and smoothness of the cross-sectional edges.
[0053] 3. Cross-section Euclidean transformation and multi-section connectivity processing.
[0054] Cross-section Euclidean transformation: Based on the design objectives, the coordinates of the control points of the tubular cross-section control polygon are transformed using Euclidean geometry. The transformation is defined as follows: in, Let Γ be the transformation matrix, Γ be the original cross section, and Γ′ be the transformed cross section. This transformation allows the cross section to generate complex irregular tubular structures along the frame line direction.
[0055] Multi-section connectivity guarantee: When multiple different cross-sections are input, it is necessary to ensure that adjacent cross-sections are connected, with a connectivity gap distance λ = 0, defined as: F(Γ1,Γ2,…,λ) = 0.5 and λ = 0. Where Γ... i This represents control polygons with different cross-sections. For control point inputs with multiple cross-sections, the coordinates of the control points must be repeatedly entered in the connected parts to ensure continuity between the cross-sections.
[0056] 4. Generate pipe surfaces based on the Frenet framework of the standard frame line.
[0057] Spatial coordinate transformation: Transform the spatial coordinates to the Frenet coordinate system space. For any point P, we have: Among them, B(s) i ), N(s i ) and T(s i ) are curves S(s) i The unit vectors of the binormal, normal, and tangent of the Frenet frame at point ) are represented by ·, where · denotes the vector dot product. This transformation allows the cross-section to be extruded along the tangent direction of the frame line, generating a complex pipe surface.
[0058] 5. Extract isosurfaces: Use the MarchingCubes algorithm to extract isosurfaces with F(x,y,z) = 0.5 and generate a mesh. Here, F(x,y,z) is a function describing the pipe structure. This algorithm converts the implicitly represented pipe structure into an explicit mesh model, facilitating subsequent visualization and processing.
[0059] 6. Use tools such as VTK to visualize the generated mesh, intuitively displaying the shape and details of complex pipe structures, so as to facilitate the inspection and modification of the design.
[0060] Furthermore, 2D piecewise algebraic spline basis functions are derived from practical observations of the energy required to bend lines in the real world, such as wires of varying radii. To bend a rigid wire into the desired shape, the wire must be long enough, or the force applied to bend it must be very large. However, for a very flexible wire, since the force applied to it is only effective over a small area of the wire, it can be easily bent to any desired curvature.
[0061] (1) 2D piecewise algebraic spline basis functions
[0062] For any given polygon Define the function as:
[0063]
[0064] Where: δ is the parameter controlling polygon smoothing. Let represent a square of size 2δ×2δ centered at the origin; the function has the following two properties: (1) It is a piecewise polynomial function. (2) It is C n-1 Continuous.
[0065] (2) 2D piecewise algebraic spline basis function explicit expression.
[0066] By introducing implicit points, implicit edges, and implicit polygons, each control polygon can be expressed as follows:
[0067]
[0068] Where V(Δ) represents the vertex controlling the polygon Δ.
[0069] Optionally, the Frenet framework method for the standard frame line in this embodiment is as follows:
[0070] Let curve S(s) i () is a parametric curve, and its tangent T, normal N, and binormal B are defined as follows:
[0071]
[0072] Transformation of spatial coordinates to Frenet coordinate system space:
[0073]
[0074] Among them, B(s) i ), N(s i ) and T(s i ) represents S(s i The unit vectors of the binormal, normal, and tangent of the Frenet frame at point ).
[0075] Furthermore, the specific method for cross-section transformation is as follows:
[0076] The transformation is defined as follows:
[0077]
[0078] in, Let Γ' represent the transformation matrix, Γ′ represent the transformed section, and Γ represent the original section.
[0079] Furthermore, the specific steps for controlling the connectivity of multi-section polygons in this embodiment are as follows:
[0080] In a two-dimensional plane, multiple cross sections are designed individually, with a connection gap distance λ = 0 between cross sections, and ensuring connectivity between adjacent cross sections.
[0081] The definition is as follows:
[0082] F(Γ1,Γ2,...,λ)=0.5andλ=0
[0083] Where Γ represents the control polygons with different cross sections, and λ represents the distance between connected gaps.
[0084] The main innovation of this invention lies in utilizing 2D piecewise algebraic splines as a method for generating complex tubular structures. By applying cross-section transformation methods and ensuring connectivity between multiple cross-sections, complex tubes are generated. Unlike traditional methods based on polygons, parametric methods, or tensor products, the 2D piecewise algebraic spline method, combined with cross-section transformation, can generate curved complex tubes along straight lines. Furthermore, by combining cross-section connectivity, it can batch generate complex tubes with different cross-sectional shapes. Specifically, this patent controls the smoothness of polygons by adjusting δ, making the edges of the cross-sections more rounded. In special cases, by adjusting a suitable δ value, the cross-section can approach a standard circle. These innovations significantly expand upon traditional methods for generating curved tubes, while also improving the efficiency of batch generating complex tubes.
[0085] like Figures 2 to 7This invention can implicitly generate complex curved tubes and can batch generate complex tubes with different cross-sectional shapes, possessing the ability to smoothly control the cross-sectional shape of complex tubes. This technology can be widely used in the modeling and design of complex industrial tubes.
[0086] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably.
[0087] This document uses specific examples to illustrate the principles and implementation methods of the present invention. The descriptions of the above embodiments are only for the purpose of helping to understand the method and core ideas of the present invention. Furthermore, those skilled in the art will recognize that, based on the ideas of the present invention, there will be changes in the specific implementation methods and application scope. Therefore, the content of this specification should not be construed as a limitation of the present invention.
Claims
1. A method for modeling complex tubular structures based on 2D piecewise algebraic splines, characterized in that, include: Select several control points in a two-dimensional plane and connect them in a counterclockwise order to form a cross-sectional control polygon; Based on the cross-sectional control polygon, a zero-order 2D piecewise algebraic spline basis function is constructed, and a higher-order basis function is constructed through integral recursion; wherein the smoothing parameter of the higher-order basis function is used to adjust the smoothness of the cross-sectional edge. A guide curve is defined in three-dimensional space, and the Frenet frame at each parameter point of the guide curve is calculated. The cross-section control polygons are mapped to the corresponding Frenet frame at each parameter point by Euclidean transformation to obtain multi-section control polygons distributed along the guide curve. Using the multi-section control polygon and the corresponding 2D piecewise algebraic spline basis function as parameters, an implicit field is constructed, and an implicit representation of the complex tubular structure is generated. The MarchingCubes algorithm is applied to the implicit field to extract isosurfaces, thereby obtaining an explicit triangular mesh model of the complex tubular structure. The triangular mesh model is then visualized, rendered, or edited.
2. The method for modeling complex tubular structures based on 2D piecewise algebraic splines according to claim 1, characterized in that, The zero-order 2D piecewise algebraic spline basis function The expression is: Where χΔ(x,y) is the characteristic function, representing that the point (x,y) controls the polygon Δ. x is the x-coordinate of a point, and y is the y-coordinate of a point.
3. The method for modeling complex tubular structures based on 2D piecewise algebraic splines according to claim 2, characterized in that, The expression for the higher-order basis functions is: Where δ is a smoothing parameter used to control the smoothness of the tubular cross-section, s is the abscissa of a point on the two-dimensional plane, t is the ordinate of a point on the two-dimensional plane, and n is the order of the spline, representing the degree of function smoothing.
4. The method for modeling complex tubular structures based on 2D piecewise algebraic splines according to claim 1, characterized in that, The Euclidean transformation process is as follows: first, rotate around the tangent direction of the Frenet frame, and then translate along the position vector of the guide curve.
5. The method for modeling complex tubular structures based on 2D piecewise algebraic splines according to claim 1, characterized in that, The connection gap distance D between adjacent cross-section control polygons is 0.
6. The method for modeling complex tubular structures based on 2D piecewise algebraic splines according to claim 1, characterized in that, Two adjacent cross-section control polygons share at least one identical control point on their common edge to ensure multi-section connectivity.
7. The method for modeling complex tubular structures based on 2D piecewise algebraic splines according to claim 1, characterized in that, After extracting isosurfaces from the implicit field using the MarchingCubes algorithm to obtain the explicit triangular mesh model of the complex tubular structure, the process further includes: The VTK library is invoked to perform smoothing, denoising, and texture mapping processing on the triangular mesh model.