A method for representing the boundary of 3D solids based on partially ordered sets of bonded compact surfaces

Abstract

This application relates to a method for representing the boundary of a 3D solid based on a partially ordered set of bonded compact surfaces, belonging to the technical field of computer graphics and computer-aided design. The method includes: inputting the 3D solid to be processed into a 3D Euclidean space, and formally defining the 3D solid as a mathematical object, an Ing set Y, defined as a bounded, regular semi-analytic open set; traversing the solid boundary of the Ing set Y, extracting singularities on the solid boundary and constructing a singularity graph structure; processing the Ing set Y and the singularity graph structure using a bonded compact surface decomposition algorithm, outputting a partially ordered set, and determining the unique boundary representation of the 3D solid based on the partially ordered set; and reading topological invariants based on the hierarchical structure of the partially ordered set in constant time complexity. This method achieves a unique representation of the 3D solid boundary, efficient querying of topological information, and complete management of boundary singularities.

Description

Technical Field

[0001] This application relates to the technical fields of computer graphics and computer-aided design, and in particular to a method for representing the boundary of three-dimensional solids based on a partially ordered set of bonded tight surfaces. Background Technology

[0002] Three-dimensional continuum modeling (solid modeling) is the digital cornerstone of engineering and science. By accurately constructing the continuous geometric and physical property representation of matter in three-dimensional space, it provides core support for the design, analysis, and verification of complex systems. In aircraft airworthiness certification, it directly serves the simulation verification of structural strength and safety performance; in automobile manufacturing, it runs through the entire process from styling design to crash simulation, significantly improving development efficiency; in the field of physical simulation, it constructs virtual experimental environments for multiphysics coupling analysis, promoting scientific computing and engineering innovation.

[0003] In 3D solid modeling, boundary representation (B-rep) is one of the mainstream techniques. However, traditional B-rep methods have significant drawbacks when dealing with solids with complex topological structures:

[0004] 1. Non-unique representation: The same entity may have multiple different boundary representations, leading to data redundancy and difficulty in maintaining consistency.

[0005] 2. Low efficiency in topology information query: Querying global topology attributes such as the connected components of an entity and the number of holes requires complex calculations, which is difficult to meet the needs of real-time interactive applications.

[0006] 3. Singularities are difficult to describe precisely: Singularities on entity boundaries (such as sharp edges and corners) lack a unified mathematical description and management mechanism, resulting in poor robustness of geometric processing algorithms (such as mesh generation and Boolean operations). Summary of the Invention

[0007] To address the aforementioned issues, this application provides a method for representing 3D solid boundaries based on a partially ordered set of bonded compact surfaces, enabling unique representation of 3D solid boundaries, efficient querying of topological information, and complete management of boundary singularities; the following technical solution is adopted:

[0008] A method for representing the boundary of a 3D solid based on a partially ordered set of bonded compact surfaces includes:

[0009] Input the three-dimensional entity to be processed into the three-dimensional Euclidean space, and formally define the three-dimensional entity as a mathematical object Yin set Y. Yin set Y is defined as a bounded regular semianalytic open set.

[0010] Traverse the entity boundary of the Yin set Y, extract the singularities on the entity boundary and construct a singularity graph structure. The singularity represents the non-manifold point or the manifold point with a non-unique tangent plane on the entity boundary.

[0011] The bounded compact surface decomposition algorithm is used to process the set Y and the singular graph structure, outputting a partially ordered set, and the unique boundary representation of the three-dimensional entity is determined based on the partially ordered set. The bounded compact surface is the result of a directional two-dimensional compact manifold being bounded along the singular graph structure in quotient space topology, satisfying that the complement in the three-dimensional Euclidean space contains two connected components: a bounded component and an unbounded component.

[0012] Read the topological invariants based on the hierarchical structure of the partially ordered set in constant time complexity.

[0013] By adopting the above technical solution, the three-dimensional entity is formalized into a mathematical object called Yin set. The entity boundary is defined by the partially ordered set output by the glued tight surface decomposition algorithm. This ensures the uniqueness of the boundary representation of the same three-dimensional entity from a mathematical perspective, completely eliminates the data redundancy problem, and reduces the data consistency maintenance cost in the entity modeling process. The constructed hierarchical structure of the partially ordered set stores the topological invariants of the entity in a structured form, which reduces the query time complexity of global topological attributes such as connected components and the number of holes to O(1), improves the query efficiency of topological information, and can fully meet the performance requirements of real-time interactive modeling applications. By traversing the entity boundary to extract singularities and constructing a singularity graph structure, a unified description and management mechanism is established for singularities on the entity boundary. This prevents sharp edges, corners, and other singularities from becoming obstacles to geometric processing algorithms, and improves the robustness of subsequent geometric processing operations such as mesh generation and Boolean operations. In summary, this technical solution, based on rigorous mathematical theory, has revolutionized traditional B-rep technology in three key dimensions: uniqueness of representation, query efficiency, and singularity handling. It can provide more efficient, accurate, and stable core support for digital modeling in fields that rely on 3D entity modeling, such as aircraft airworthiness certification, automobile manufacturing, and physical simulation.

[0014] Optionally, the three-dimensional solid boundary representation method further includes:

[0015] The extracted singularities are divided into two categories: isolated singularities and non-isolated singularities. Isolated singularities are characterized by the fact that the intersection of the small neighborhood corresponding to the singularity and the actual boundary of the Yin set Y contains only the singularity itself. Non-isolated singularities are characterized by the continuous distribution of singularities to form a singularity curve, including vertices and non-vertices. Vertices are characterized by the endpoints of a boundary curve or the points where at least three boundary curves intersect. Non-vertices are characterized by points on the singularity curve other than vertices.

[0016] Optionally, the specific steps of traversing the entity boundaries of the Yin set Y, extracting singularities on the entity boundaries, and constructing a singularity graph structure include:

[0017] Treat each singularity of vertex type as a vertex of the graph. For each continuous curve consisting of non-vertex singularities, connect the two vertices at its two ends with an edge and mark the isolated singularity.

[0018] Output a one-dimensional CW complex, i.e., an undirected graph G=(V,E), where V is the set of vertices and E is the set of edges.

[0019] By adopting the above technical solution, singularities are meticulously divided into isolated singularities and non-isolated singularities. Non-isolated singularities are further distinguished into vertices and non-vertices, achieving precise recognition and differentiated management of various singularities on entity boundaries. This breaks through the predicament of vague singularity descriptions and disordered management in traditional B-rep methods, providing clear classification standards and definition boundaries for previously difficult-to-handle singularities such as sharp edges and corners. Based on this, a graph structure construction logic using vertex-type singularities as graph vertices and continuous curves formed by non-vertex singularities as connecting edges transforms the complex singularity distribution relationships into an intuitive one-dimensional CW complex undirected graph. This not only makes the topological relationships between singularities clearly visible but also structurally integrates scattered singularity information, reducing the difficulty of analyzing the global distribution and interrelationships of singularities. This structured graph structure also provides clear topological guidance for subsequent glued tight surface decomposition algorithms, which can help the algorithm more efficiently identify the topological features of entity boundaries, making the boundary representation of 3D entities unique. The labeling of isolated singularities also ensures that every special point can be included in a unified management system, avoiding the problem of decreased geometric robustness caused by the omission of singularities.

[0020] Optionally, the specific steps for processing the Yin set Y and the singular graph structure using the glued tight surface decomposition algorithm to output the partially ordered set include:

[0021] According to the singularity graph structure, the entity boundary of the Yin set Y is decomposed into an initial set of surface patches. Each surface patch in the initial set of surface patches is a surface with edges, and its boundary is composed of smooth curve segments and / or boundary segments containing isolated singularities.

[0022] The initial set of surface patches is paired and glued to obtain a set of closed surfaces;

[0023] The closed surface set is then paired and bonded again to obtain bonded tight surfaces, which include positively oriented bonded tight surfaces and negatively oriented bonded tight surfaces.

[0024] Add the bonded surfaces to the final result set and check if the set of closed surfaces is empty;

[0025] If so, then output the partial order set based on the final result set and the predefined partial order relation.

[0026] By adopting the above technical solution, the initial surface patch decomposition of the Yinji entity boundary based on the singularity graph structure decomposes the complex entity boundary into regular edge-bounded surface patches. These surface patches are bounded by smooth curve segments and boundary segments containing isolated singularities, which not only preserves all the geometric and topological information of the entity boundary, but also makes the originally complex boundary structure modular and orderly. Through two rounds of pairing and gluing operations, a closed surface set is gradually generated from the initial surface patch set until a tightly glued surface is formed. This process realizes the gradual regularization and simplification of the topological structure, and organically integrates the scattered surface patches through topological equivalence relations. This not only eliminates redundant boundary information, but also constructs a compact surface structure that meets the requirements of topological consistency. This transforms the originally complex three-dimensional entity boundary into a compact surface set with clear topological attributes, improving the simplicity and accuracy of the three-dimensional entity topological representation. The generated partial-order set hierarchical structure stores the topological attributes of the entity in a structured form, reducing the query time complexity of global topological attributes such as connected components and the number of holes to O(1), which meets the high-efficiency requirements of real-time interactive applications.

[0027] Optionally, the specific steps for decomposing the solid boundary of the Yin set Y into an initial set of surface patches according to the singularity graph structure include:

[0028] For each edge in the singularity graph structure, the solid boundary of the Yin set Y is cut along the edge with more than 2 surface patches attached.

[0029] For each isolated singularity, a local spherical cut is made around it to isolate itself from the continuous boundary;

[0030] After processing the singularity graph structure, an initial set of surface patches is output.

[0031] By employing the aforementioned technical solutions, targeted cutting of edges with more than two surface patches attached to a singularity graph can identify and segment complex topological connection regions on the entity boundary. The originally intertwined surface structures are clearly decomposed, preserving the geometric features of each surface patch while clarifying their original connections, thus avoiding the loss of topological information that may occur in traditional decomposition methods. For local spherical cutting of isolated singularities, independent spherical boundaries are constructed around the singularity, precisely isolating these difficult-to-handle singularities from continuous boundaries. This ensures the complete preservation of the geometric features of isolated singularities while preventing them from interfering with subsequent surface patch bonding and topological lookup operations, solving the core pain points of ambiguous singularity descriptions and poor robustness in traditional B-rep methods. The initial set of surface patches output after these two targeted cutting processes has clear boundary definitions and independent topological attributes for each surface patch, completely covering the entire boundary region of the original 3D entity without any redundant or overlapping parts.

[0032] Optionally, the steps following the decomposition into an initial set of surface patches include:

[0033] Examine each element in the initial set of surface patches. If a surface patch has no boundary, it is determined to be a tightly bonded surface.

[0034] The surface patches characterized as tightly bonded surfaces are removed from the initial set of surface patches and moved into the final result set.

[0035] By adopting the above technical solution, the boundary features of each initial surface patch are checked, and the surface patches that already meet the characteristics of tightly bonded surfaces can be identified. These surface patches do not need to participate in the subsequent complex pairing and bonding operations and are directly moved into the final result set. This avoids unnecessary computational consumption, reduces possible operational errors in the intermediate process, and improves the overall efficiency of the algorithm.

[0036] Optionally, the specific steps for pairing and bonding the initial set of surface patches to obtain a closed set of surfaces include:

[0037] Select a surface patch from the initial set of surface patches as the starting surface, and locate a boundary curve of the starting surface;

[0038] Find another surface patch that shares the boundary curve and is located on the other side of the boundary curve, and pair it with the starting surface to form the boundary of the local interior of the three-dimensional solid;

[0039] Two surface patches are topologically bonded along the boundary curve to form a bonded surface combination. The bonded surface combination is then used as the processing object to pair and bond the remaining boundary with other surface patches until the formed surface meets the requirements of a closed surface. Finally, the surface patches that participated in the construction of the surface are removed from the initial set of surface patches.

[0040] The remaining surface patches continue to repeat the pairing and bonding process until there are no more surface patches available for pairing in the initial set of surface patches.

[0041] By adopting the above technical solution, the logic of starting with a single surface patch and pairing based on shared boundary curves can identify the internal topological relationships between surface patches. This ensures that each bonding operation is based on the real boundary relationships within the entity, avoiding meaningless pairing attempts and improving the targeting and accuracy of the bonding process. The iterative bonding method, which treats the combination of bonded surfaces as the processing object, can gradually expand the coverage of internal closed surfaces, integrating scattered surface patches into complete internally bonded tight surfaces. This ensures the topological correctness of each bonding operation and verifies in real time whether the integrated surface meets the topological closure requirements, ensuring that each generated internally bonded tight surface has a complete and self-consistent topological structure. This fundamentally eliminates the boundary representation inconsistency problem that may occur in the traditional B-rep method. The iterative processing mode can comprehensively traverse all pairable elements in the initial surface patch set, ensuring that all surface patches that can form internal closed boundaries are fully integrated, reducing redundant data in subsequent processing, and also allowing the complete extraction and structured storage of the internal topological structure of the 3D entity.

[0042] Optionally, the specific steps for performing pairing and bonding processing on the closed surface set again to obtain bonded tight surfaces include:

[0043] The surface patches in the closed surface set are cut twice, and the cutting condition is that the surface patches connect at least three spatially non-adjacent external regions.

[0044] Select one of the surface patches as the initial surface, and locate a boundary curve of the initial surface;

[0045] Find another surface patch that shares the boundary curve and is located on the other side of the boundary curve, and pair it with the initial surface to form the external boundary of the local three-dimensional solid.

[0046] Two surface patches are topologically bonded along the boundary curve to form a new surface combination. The new surface combination is then used as the processing object to pair and bond the remaining boundary with other surface patches until the formed surface meets the requirements of tightly bonded surfaces. Finally, the surface patches that participated in the construction of the surface are removed from the closed surface set.

[0047] The remaining surface patches continue to repeat the pairing and bonding process until the closed surface set no longer has surface patches available for pairing.

[0048] By employing the aforementioned technical solution, the initial secondary cutting step can identify and separate surface patches that connect multiple non-adjacent external regions, solving the problem of subsequent pairing and bonding difficulties caused by the complex topological structure of surface patches, making the originally intertwined external boundary relationships clearly discernible. The pairing logic, using the boundary curves of the surface patches as clues, can locate corresponding surface patches sharing a boundary, ensuring that each pairing is based on the real topological association outside the entity, avoiding boundary representation distortion caused by incorrect pairing. Through multiple iterations of bonding until a tightly bonded surface meets the requirements, scattered surface patches can be gradually integrated into complete external boundary units, fully restoring the external topological morphology of the 3D entity. The iterative processing mode comprehensively covers all pairable surface patches, ensuring that all surface patches that can form external closed boundaries are fully integrated, reducing redundant data while allowing the complete extraction and structured storage of the external topological structure of the 3D entity. The finally generated external tightly bonded surface, together with the previously obtained internal tightly bonded surface, constitutes the complete boundary topology set of the 3D entity.

[0049] Optionally, the partial order relation defines inclusion and coverage relations; wherein, the inclusion relation is defined as follows: if and only if the surfaces in the final result set... The bounded region enclosed by the surface is completely contained within the surface. Within the bounded region enclosed by the boundary, the covering relationship is defined as if And there is no other surface. Make Then it is called cover This indicates a direct inclusion relationship.

[0050] By adopting the above technical solutions, the definition of inclusion relationship takes the bounded region enclosed by the surface as the core criterion, which can accurately characterize the spatial nesting relationship between different tightly bonded surfaces, transforming the complex topological structure of 3D entities into a clear hierarchical logic, making the originally abstract spatial positional relationship quantifiable and determinable. The definition of covering relationship, based on inclusion relationship, further clarifies the direct nesting association between surfaces, eliminates the indirect influence of intermediate levels, and forms a set of non-redundant direct topological association network. This design provides an efficient path for subsequent topological query operations. When it is necessary to query all nested sub-surfaces or direct parent surfaces of a certain surface, it is not necessary to traverse the entire topological structure. It is only necessary to rely on the direct association network constructed by covering relationship to quickly locate it, reducing the query time complexity of topological invariants such as connected components and the number of internal holes to O(1), and improving the efficiency of topological query. The definition of partial order relationship also makes the boundary representation of 3D entities unique. No matter how the initial surface patch cutting order changes, the final generated partial order set can reflect the real topological structure of the entity, solving the core pain point of non-unique boundary representation in traditional methods.

[0051] Optionally, the topological invariants include the number of connected components and the number of genus; the number of connected components is equal to the number of positively oriented, tightly bound surfaces, and the number of genus is derived within the time complexity based on the number, type, and coverage relationship of nodes in each branch of the partial order set using Euler's characteristic formula.

[0052] By adopting the above technical solution, the number of connected components is bound to the number of surfaces without parent node coverage in the partial order set. This eliminates the need for global traversal analysis of the entire 3D entity's boundary; results can be quickly obtained by simply counting the top-level nodes of the partial order set. This design directly reduces the query time complexity of connected components to O(1), transforming the originally complex global topology analysis into a simple node counting operation, saving computational resources. The derivation of the genus number relies on the number, type, and coverage relationship of nodes in each branch of the partial order set, combined with Euler's characteristic formula. This method fully utilizes the structured topological association information already stored in the partial order set, avoiding the tedious process of reconstructing topological relationships in traditional methods, and also achieving efficient topology queries. This method of calculating topological invariants based on partial order sets relies entirely on the already structured topological relationship data, without depending on the initial surface patch cutting order or intermediate processing, ensuring the uniqueness and accuracy of the topological invariant results and solving the problem of inconsistent topological attribute query results caused by non-unique boundary representations in the traditional B-rep method.

[0053] In summary, this application includes at least one of the following beneficial technical effects:

[0054] 1. Uniqueness and efficiency of representation. By using the partially ordered set of tightly bound surfaces as the core data structure, a one-to-one correspondence between 3D entities and their boundary representations is guaranteed, eliminating the ambiguity caused by data redundancy in traditional methods, while also reducing storage space.

[0055] 2. Extremely fast topology query capability. By leveraging the hierarchical characteristics of partially ordered sets, it achieves indexing of entity topology information in constant time complexity, breaking through the computational bottleneck of traditional boundary representation methods when querying topology information, and greatly improving efficiency.

[0056] 3. A complete singularity management mechanism. Organizing all singularities on the entity boundary using a graph structure not only solves the problem of accurately representing complex entity models (such as those containing non-manifold boundaries), but also provides a more robust geometric foundation for subsequent applications such as mesh generation and physical simulation. Attached Figure Description

[0057] Figure 1 This is a first flowchart of an embodiment of the method of this application;

[0058] Figure 2 This is a second flowchart of an embodiment of the method of this application;

[0059] Figure 3 This is a third flowchart of an embodiment of the method of this application;

[0060] Figure 4 This is the fourth flowchart of an embodiment of the method of this application;

[0061] Figure 5 This is the fifth flowchart of an embodiment of the method of this application;

[0062] Figure 6 It is a partial example of a partially ordered set composed of tightly bonded surfaces. Detailed Implementation

[0063] To make the purpose, technical solution, and advantages of this application clearer, the following description is provided in conjunction with the appendix. Figures 1-6 The present application will be further described in detail below with reference to embodiments. It should be understood that the specific embodiments described herein are for illustrative purposes only and are not intended to limit the scope of the application.

[0064] This application discloses a method for representing the boundary of a three-dimensional solid based on a partially ordered set of bonded tight surfaces. (Refer to...) Figure 1 The three-dimensional solid boundary representation method includes S110-S140:

[0065] S110, input the three-dimensional entity to be processed into the three-dimensional Euclidean space, and formally define the three-dimensional entity as a mathematical object Yin set Y. The Yin set Y is defined as a bounded regular semianalytic open set.

[0066] S120, traverse the entity boundary of Yin set Y, extract the singularities on the entity boundary and construct the singularity graph structure. The singularity represents the non-manifold point or the manifold point with a non-unique tangent plane on the entity boundary.

[0067] S130 uses the glued compact surface decomposition algorithm to process the Yin set Y and the singular graph structure, outputs the partially ordered set, and determines the unique boundary representation of the three-dimensional entity based on the partially ordered set; the glued compact surface is the result of a directional two-dimensional compact manifold being glued along the singular graph structure in quotient space topology, which satisfies that the complement in three-dimensional Euclidean space contains two connected components, a bounded component and an unbounded component.

[0068] S140 reads topological invariants based on the hierarchical structure of a partially ordered set in constant time complexity.

[0069] Specifically, in step S110, the input 3D entity is considered to be embedded in 3D Euclidean space ( An object within a set. For precise mathematical analysis (such as boundary extraction, singularity identification, and topological operations), the entity is formally defined as a "Yin Set" Y. Formalizing a 3D entity as a Yin Set Y is a rigorous modeling paradigm based on semi-analytic set theory and regularity theory. "Yin Set" is an original term in this scheme, essentially a reinforcement of the classic "regular open set": requiring Y to not only be an open set, but also that the interior of its closure is exactly itself (…). ), and the entity boundary of Y The set must be semi-analytical, meaning it can be locally constructed from the zeros, positive sets, or negative sets of a finite number of real analytical functions through Boolean operations and projection. This condition ensures that the boundary has a locally finite singularity structure, providing an analytical guarantee for subsequent singularity classification. In practice, if the input entity comes from parametric surfaces (NURBS), implicit functions (such as RBF, MPU), voxel meshes, or CSG trees, it needs to be uniformly converted into a semi-analytical approximation model with accuracy guarantees. For example, for NURBS entities, a high-order implicit approximation F(x,y,z)=0 is generated using T-mesh adaptive subdivision and Greville point interpolation, and the regularity is verified by sampling with the signed distance field (SDF). For voxel data, the Marching Cubes++ algorithm combined with topological verification (such as critical point analysis based on Morse theory) is used to extract the boundary mesh that satisfies the semi-analytical constraints, and local polynomial fitting (such as Moving Least Squares) is used to improve the resolution. The output Y in this step is not only a set, but also a queryable computational object that supports low-level operations such as boundary point sampling, normal consistency determination, and neighborhood homeomorphism testing.

[0070] The specific steps of S120 include:

[0071] Traversing the entity boundary of Yinji Y, the extracted singularities are divided into two categories: isolated singularities and non-isolated singularities. Isolated singularities are characterized by the fact that the intersection of the small neighborhood corresponding to the singularity and the actual boundary of Yinji Y contains only the singularity itself. Non-isolated singularities are characterized by the continuous distribution of singularities to form a singularity curve, including vertices and non-vertices. Vertices are characterized by the endpoints of a boundary curve or the points where at least three boundary curves intersect. Non-vertices are characterized by points on the singularity curve other than vertices.

[0072] Then, each singularity of vertex type is treated as a vertex of the graph. For each continuous curve composed of non-vertex singularities, the two vertices connected at its two ends are connected by an edge, and isolated singularities are specially marked. The final output is a one-dimensional CW complex, i.e., an undirected graph G=(V,E), where V is the vertex set and E is the edge set.

[0073] Specifically, in step S120, singularity classification is the topological awareness center of the entire scheme. A singularity is defined as a point in the entity boundary that does not satisfy the local homeomorphism condition of a two-dimensional manifold, i.e., it does not have a neighborhood. Make Homeomorphism Based on their neighborhood topological behavior, singularities are divided into isolated singularities and non-isolated singularities.

[0074] To obtain the candidate point set of the singularity, Perform multiscale curvature tensor analysis: calculate principal curvatures and its derivative, define the singularity indicator function. ,exist Initiate a high-precision critical point search (such as Levenberg-Marquardt optimization) for candidate point sets in the local maximum region.

[0075] For each candidate point p, a sphere is constructed with a sufficiently small radius r centered on p. Calculate the boundary between the sphere and the solid. intersection If there is only one singularity candidate point p in the intersection, it is determined to be an isolated singularity; otherwise, it is a non-isolated singularity.

[0076] Non-isolated singularities further distinguish between vertices and non-vertices: a vertex is an endpoint of a boundary curve or requires its connection to another point. A connection must have at least three connected components (corresponding to the intersection of three or more boundary curves), which can be achieved by calculating the degree of the nodes in the Reeb graph of the link. A non-vertex connection refers to a single-branch connection that is not circular (e.g., a figure-eight shape, corresponding to a cusp on the edge). In this case, the Whitney layering algorithm (Whitney's Condition B) must be called to verify that it belongs to a 1D layer. All singularity detection is performed in the parameter domain or implicit gradient domain to avoid grid discretization errors. For example, for implicit models, this is achieved by tracking... The critical point set and its union Find the intersection, and then use Hessian matrix sign eigenvalue analysis to determine the type of singularity.

[0077] Vertex set V corresponds to all identified vertex-type singularities; the construction of edge set E requires singularity curve tracing: for each non-vertex singularity, integrate the manifold along its layer direction, i.e., solve the differential equation. ,in for The singular orientation field at x is jointly determined by the Hessian null space of F and the gradient orthogonal complement space until another vertex is reached or the field closes. This process employs an implicit curve tracker with an adaptive step size (such as Pseudo-Arc-LengthContinuation) and verifies Whitney layer consistency in real time. Isolated singularities do not participate in edge construction but are treated as dangling vertices with special markings (such as...). The existence of a graph structure itself represents local nonmanifoldness (such as a spherical perforation point). The final singularity graph structure G is not only a graph, but also... The skeleton carries the coarsest-grained topological connection information of the entity. Its data structure adopts a half-edge structure with attributes, which supports O(1) adjacency query and manifold tracing.

[0078] It should be noted that, in other implementations, the singularity graph structure can be directly input by the user or specified by the user directly from a stored singularity graph structure.

[0079] Reference Figure 2 In S130, the specific steps for processing the Yin set Y and the singular graph structure using the glued tight surface decomposition algorithm to output the partially ordered set include S210-S250:

[0080] S210, according to the singularity graph structure, the entity boundary of the Yin set Y is decomposed into an initial set of surface patches. Each surface patch in the initial set of surface patches is a surface with edges, and its boundary is composed of smooth curve segments and / or boundary segments containing isolated singularities.

[0081] S220, the initial set of surface patches is paired and glued to obtain a set of closed surfaces;

[0082] S230, the closed surface set is paired and glued again to obtain a glued tight surface;

[0083] S240, add the bonded surfaces to the final result set and check if the closed surface set is empty;

[0084] If the result is S250, then output the partial order set based on the final result set and the predefined partial order relation; otherwise, return to S220 or S230 to continue iterative processing.

[0085] Reference Figure 3 The specific steps of S210 include S310-S330:

[0086] S310, For each edge in the singular graph structure, cut the solid boundary of the Yin set Y along the edge with more than 2 surface patches attached.

[0087] S320, for each isolated singularity, performs a local spherical cut around it to isolate itself from the continuous boundary;

[0088] S330, after processing the singularity graph structure, outputs the initial set of surface patches.

[0089] Specifically, steps S310 and S320 together constitute the singularity-guided boundary cutting core. S310 processes edges with more than two surface patches attached, specifically the edges corresponding to vertices v in G with degree deg(v) ≥ 3. These edges... The above represents the common boundary of multiple facets (such as an edge where three faces meet). The cutting operation is essentially a topological splitting with normal displacement along that edge: the edge e is parameterized as... , Calculate its unit normal field n(t), which is obtained by normalizing the cross product of the normal fields of the two side panels, and then generate two offset curves: and , Take it as 1 / 10 of the local radius of curvature; then use these two curves as the toolpath, in... The implicit cut based on the level set is performed to ensure that the new boundary is smooth and topologically preserved.

[0090] S320 performs a local spherical cut on the isolated singularity q: constructing a sphere centered at q with radius z. ,calculate We obtain a closed curve c; then we use c as the new boundary, and... The surface is divided into two parts along c. The outer part remains connected, while the inner part forms a "singularity capsule" centered at q. This capsule is isolated as a degenerate surface patch with c as its boundary. This operation is equivalent to collapsing q into a point in the quotient space, laying the foundation for subsequent bonding.

[0091] The initial set of surface patches output by S330 serves as the intermediate computational carrier for the entire scheme; each element... It is a compact surface with edges, its boundary It is composed of several smooth arc segments and several isolated singularity segments (i.e., the capsule boundary c generated by S320). Here, "edge" refers to... It is a two-dimensional manifold, but Not empty; compactness ensures it can be embedded in a compact space for easy computation. In implementation, each It is stored as parametric or implicit surface patches and maintains a boundary loop linked list. Each node contains: arc type identifier (smooth / singular), parameter range, adjacent vertex index, and edge ID in the singular graph G to which it belongs. This structure supports O(1) boundary traversal and fast lookup of adjacent patches.

[0092] Furthermore, the steps between S210 and S220 include S211-S212 (not shown in the diagram):

[0093] S211, check each element in the initial set of surface patches. If a surface patch has no boundary, it is determined to be a tightly bonded surface.

[0094] S212 removes the surface patch characterized as a tightly bonded surface from the initial set of surface patches and moves it into the final result set.

[0095] Specifically, at the geometric level, the algorithm processes each surface patch... Perform a nullability check on the bounding loop linked list: if the length of the linked list is zero (i.e., ... If the list is empty, it is initially marked as a candidate without boundaries. Then, for all marked candidate surface patches, the critical point analysis module based on Morse theory is called to calculate the degree of its Gaussian mapping on the unit sphere, and the Euler characteristic is inverted by combining the Poincaré–Hopf index theorem. If and only if The numbers are even and their absolute values ​​satisfy the following conditions: (Possible values ​​for closed surfaces such as spheres, tori, and double tori), and its first homology group. Only when the rank of the surface patch is even after Hodge decomposition (ensuring orientability) does it proceed to the final confirmation stage. The third layer focuses on verifying the realism of the 3D embedding: using the parametric representation or implicit representation of the symbolic distance field (SDF) of the surface patch, its bounding box hierarchy tree (BVH) is constructed, and self-collision detection is performed. This not only checks whether there is overlap of parameter domains within the surface patch, but also eliminates false closure cases (such as self-intersecting Klein bottle embeddings or singular intrusions with spikes) by emitting dense rays along random directions and statistically analyzing the parity stability of the number of intersections with the surface. Only when all three checks pass is the system removed from the initial surface patch set and injected into the final result set.

[0096] Reference Figure 4 The specific steps of S220 include S410-S440:

[0097] S410: Select a surface patch from the initial set of surface patches as the starting surface, and locate a boundary curve of the starting surface;

[0098] S420: Find another surface patch that shares the boundary curve and is located on the other side of the boundary curve, and pair it with the starting surface to form the boundary of the local interior of the three-dimensional solid.

[0099] S430, topologically bond the two surface patches along the boundary curve to form a bonded surface combination, and use the bonded surface combination as the processing object to pair and bond the remaining boundary with other surface patches until the formed surface meets the requirements of a closed surface, and remove the surface patches that participated in the construction of the surface from the initial surface patch set;

[0100] S440, the remaining surface patches continue to repeat the pairing and bonding process until there are no more surface patches available for pairing in the initial set of surface patches.

[0101] Specifically, at the geometric level, the algorithm processes each surface patch... Perform a pairing query on the boundary ring linked list. Select the starting face. Next, locate its boundary curve c, and find another surface patch by querying the boundary loop linked list of all surface patches. Also, c is the boundary (i.e., c is...) and The two surfaces are bound together along the common edge of c, and the two surfaces are located on opposite sides of c; the two surfaces are bonded together in quotient space along c: define an equivalence relation such that every point on c... exist and Corresponding points in the middle are equivalent, and the remaining points are self-equivalent; the resulting quotient space It remains a surface with edges, but the boundaries are reduced. The newly formed surface combination is immediately used as the current processing object, continuing to find pairs and glue them with other surface patches in the set. This is a progressively closing process. This process is repeated, and when a new surface satisfies the boundary-free condition after a certain glue, a closed surface is formed. Then, using the ray casting method combined with the sign change count of the sign distance field, the parity of the number of intersection points between rays originating from a point outside S and S is counted. If the number is always even, then S is an orientable closed surface. Then, combined with the embedding test, the image of S is calculated... Whether the surfaces are self-intersecting (detected by BBox overlap between surface patches and parameter domain conflict). If there is no self-intersection and the surface can be oriented, then the closed surface is a negatively oriented, tightly bound surface. This process essentially involves performing constrained graph matching on the initial set of surface patches, where the constraints are boundary sharing and orientation compatibility (normal pointing inwards to the Y direction), and it deals with the solid cavity and inner shell.

[0102] Reference Figure 5 The specific steps of S230 include S510-S550:

[0103] S510, perform secondary cutting on the surface patch of the closed surface set, the cutting condition is that the surface patch connects at least three spatially non-adjacent external regions;

[0104] S520, select one of the surface patches as the initial surface, and locate a boundary curve of the initial surface;

[0105] S530: Find another surface patch that shares the boundary curve and is located on the other side of the boundary curve, and pair it with the initial surface to form the external boundary of the local three-dimensional solid.

[0106] S540, topologically bond two surface patches along the boundary curve to form a new surface combination, and use the new surface combination as the processing object to pair and bond the remaining boundary with other surface patches until the formed surface meets the requirements of tightly bonded surface, and remove the surface patches that participated in the construction of the surface from the closed surface set;

[0107] S550, the remaining surface patches continue to repeat the pairing and bonding process until there are no more surface patches available for pairing in the closed surface set.

[0108] Specifically, the secondary cutting in step S510 targets the surface patch connecting multiple external regions, i.e., its boundary curve c will... The system is divided into multiple unbounded regions. The cutting condition is determined using planar segmentation topological coding: project c onto a coordinate plane, construct its planar embedded DCEL (Doubly Connected Edge List), and calculate the number of faces f; if f>3, then c is a multi-connected boundary and needs to be cut along a chord of c.

[0109] The pairing logic for subsequent S520-S550 is similar to the internal logic, gradually expanding from pairing and bonding at a single boundary to the boundary processing of the entire surface, ensuring that the bonding process between surface patches always follows the principle of topological consistency. Newly formed surface combinations are also immediately treated as the current processing objects, continuing to search for pairs and bond them. When the generated closed surface satisfies the definition of a tightly bonded surface (its complement...), the bonding process continues. When a surface contains exactly two connected components in three-dimensional space, it is removed from the closed surface set to complete the construction of the bonded surface.

[0110] If, after two surface patches are bonded along c, their normals both point outward from Y, and the bounded components of the final closed surface are empty (i.e., the region enclosed by S is entirely unbounded), then it is a positively oriented bonded tight surface. This involves handling the shell and convex hull hierarchy of entities. This can be achieved through an infinity point inclusion check: adding virtual points. Inverse inference based on the parity of ray intersections Is it in The interior.

[0111] In step S240, the final result set Includes all negatively and positively oriented bonded surfaces, initial surface patch set If, after multiple bonding operations, there are remaining surface patches that cannot be paired, it indicates a topological defect, such as incomplete cutting of non-manifold edges or missing singularities. A defect diagnosis protocol needs to be activated: For the remaining surface patches, calculate the Alexander polynomial or Jones polynomial approximation of their boundary curves (used to detect knots). If non-trivial knots are found, it indicates an error in the singularity graph structure G, requiring a backtracking to S120 to enhance singularity detection sensitivity; if the boundary is a trivial ring but cannot be paired, then adaptive [mechanism] should be enabled. Adaptive neighborhood expansion (neighborhood inflation) The remaining surface patches are slightly displaced in the normal direction to force the generation of new intersection lines to trigger a new round of cutting, and then return to S220 / S230 to continue iterative processing.

[0112] In step S250, the partial order relation defines the inclusion relation and the covering relation; where the inclusion relation ≤ definition For if and only if the surface in the final result set is The bounded area enclosed Completely contained within the surface The bounded area enclosed Inside, and Let be any two surfaces in the final result set. This inclusion relation can be transformed into an implicit function hierarchy comparison: if Depend on definition, Depend on Defined, then If the implication holds for all x, it is verified by the polynomial nonnegativity decision (SOS decomposition) or the sample-validate-counterexample algorithm.

[0113] Coverage relationship <: if And there is no other surface. Make Then it is called cover This indicates a direct containment relationship. It is constructed using a Hasse diagram: for each pair... Check if it exists If not, then This represents an edge in the Hasse graph. The final output Hasse graph is the topological fingerprint of the entity, where nodes are tightly bound surfaces and edges are directly contained.

[0114] The final result set together with the partial order relation defined on it This constitutes a finite partially ordered set. The structure of a partially ordered set can be intuitively represented as a tree diagram, such as... Figure 6 As shown, child nodes are directly covered (contained) by their parent nodes. (Yinji) It can be uniquely represented as That is, the union of all tightly bonded curved surfaces.

[0115] Finally, in step S140, the topological invariants include the number of connected components and the number of genus (holes); the number of connected components is equal to the number of positively oriented, tightly bound surfaces (i.e., The number of elements), each maximal element corresponds to an independent connected component of the entity; the number of genus elements depends on the number and type of nodes in each branch of the partial order set. The covering relationship is derived within the time complexity based on Euler's characteristic formula.

[0116] The calculation of the genus number g reuses the Euler characteristic. For each tightly bonded surface ,in, (Vertex-Edge-Face Count), and (Orientable closed surface). Inversely deduced from the partial order structure: Let... In a Hasse graph, if a subtree contains k nodes, If it is an externally bonded curved surface, then its The number of its direct child nodes, m, is determined (since each child node represents a "hole" to be filled), and the formula is: If it is an internally bonded curved surface, then Refers to the internally bonded curved surface All direct child nodes (by) The sum of genus of the covered surface. This derivation only involves a local traversal of the Hasse graph, visiting each node once, with a total time complexity of O(1). After the final result set is fixed, all topological invariants can be returned in O(1) by table lookup and simple arithmetic, truly realizing real-time computation of topology awareness.

[0117] Although this application has been described herein in conjunction with various embodiments, those skilled in the art, by reviewing the accompanying drawings, disclosure, and appended claims, will understand and implement other variations of the disclosed embodiments in carrying out the claimed application. In the claims, the word "comprising" does not exclude other components or steps. While different dependent claims may recite certain measures, this does not mean that these measures cannot be combined to produce a good effect.

[0118] The above are all preferred embodiments of this application and are not intended to limit the scope of protection of this application. Any feature disclosed in this specification (including the abstract and drawings) may be replaced by other equivalent or similar features unless specifically stated otherwise. That is, unless specifically stated otherwise, each feature is only one example of a series of equivalent or similar features.

Claims

1. A method for representing the boundary of a three-dimensional solid based on a tight quasilinear surface partial order set, characterized in that, include: Input the three-dimensional entity to be processed into the three-dimensional Euclidean space, and formally define the three-dimensional entity as a mathematical object Yin set Y. Yin set Y is defined as a bounded regular semianalytic open set. Traverse the entity boundary of the Yin set Y, extract the singularities on the entity boundary and construct a singularity graph structure. The singularities represent non-manifold points or manifold points with non-unique tangent planes on the entity boundary. The bounded compact surface decomposition algorithm is used to process the set Y and the singular graph structure, outputting a partially ordered set, and the unique boundary representation of the three-dimensional entity is determined based on the partially ordered set. The bounded compact surface is the result of a directional two-dimensional compact manifold being bounded along the singular graph structure in quotient space topology, satisfying that the complement in the three-dimensional Euclidean space contains two connected components: a bounded component and an unbounded component. Read the topological invariants based on the hierarchical structure of the partially ordered set in constant time complexity.

2. The method of claim 1, wherein, The three-dimensional solid boundary representation method also includes: The extracted singularities are divided into two categories: isolated singularities and non-isolated singularities. Isolated singularities are characterized by the fact that the intersection of the small neighborhood corresponding to the singularity and the actual boundary of the Yin set Y contains only the singularity itself. Non-isolated singularities are characterized by the continuous distribution of singularities to form a singularity curve, including vertices and non-vertices. Vertices are characterized by the endpoints of a boundary curve or the points where at least three boundary curves intersect. Non-vertices are characterized by points on the singularity curve other than vertices.

3. The method of claim 2, wherein, The specific steps for traversing the entity boundaries of the Yin set Y, extracting singularities on the entity boundaries, and constructing a singularity graph structure include: Treat each singularity of vertex type as a vertex of the graph. For each continuous curve consisting of non-vertex singularities, connect the two vertices at its two ends with an edge and mark the isolated singularity. Output a one-dimensional CW complex, i.e., an undirected graph G=(V,E), where V is the set of vertices and E is the set of edges.

4. The method for representing the boundary of a three-dimensional solid based on a partially ordered set of bonded tight surfaces according to claim 1, characterized in that, The specific steps for processing the Yin set Y and the singular graph structure using the glued tight surface decomposition algorithm to output the partially ordered set include: According to the singularity graph structure, the entity boundary of the Yin set Y is decomposed into an initial set of surface patches. Each surface patch in the initial set of surface patches is a surface with edges, and its boundary is composed of smooth curve segments and / or boundary segments containing isolated singularities. The initial set of surface patches is paired and glued to obtain a set of closed surfaces; The closed surface set is then paired and bonded again to obtain bonded tight surfaces, which include positively oriented bonded tight surfaces and negatively oriented bonded tight surfaces. Add the bonded surfaces to the final result set and check if the set of closed surfaces is empty; If so, then output the partial order set based on the final result set and the predefined partial order relation.

5. The method for representing the boundary of a three-dimensional solid based on a partially ordered set of bonded tight surfaces according to claim 4, characterized in that, The specific steps for decomposing the solid boundary of the Yin set Y into an initial set of surface patches based on the singularity graph structure include: For each edge in the singularity graph structure, the solid boundary of the Yin set Y is cut along the edge with more than 2 surface patches attached. For each isolated singularity, a local spherical cut is made around it to isolate itself from the continuous boundary; After processing the singularity graph structure, an initial set of surface patches is output.

6. The method for representing the boundary of a three-dimensional solid based on a partially ordered set of bonded tight surfaces according to claim 4, characterized in that, The steps following the decomposition into an initial set of surface patches include: Examine each element in the initial set of surface patches. If a surface patch has no boundary, it is determined to be a tightly bonded surface. The surface patches characterized as tightly bonded surfaces are removed from the initial set of surface patches and moved into the final result set.

7. The method for representing the boundary of a three-dimensional solid based on a partially ordered set of bonded tight surfaces according to claim 4, characterized in that, The specific steps for pairing and bonding the initial set of surface patches to obtain a closed set of surfaces include: Select a surface patch from the initial set of surface patches as the starting surface, and locate a boundary curve of the starting surface; Find another surface patch that shares the boundary curve and is located on the other side of the boundary curve, and pair it with the starting surface to form the boundary of the local interior of the three-dimensional solid; Two surface patches are topologically bonded along the boundary curve to form a bonded surface combination. The bonded surface combination is then used as the processing object to pair and bond the remaining boundary with other surface patches until the formed surface meets the requirements of a closed surface. Finally, the surface patches that participated in the construction of the surface are removed from the initial set of surface patches. The remaining surface patches continue to repeat the pairing and bonding process until there are no more surface patches available for pairing in the initial set of surface patches.

8. The method for representing the boundary of a three-dimensional solid based on a partially ordered set of bonded tight surfaces according to claim 7, characterized in that, The specific steps for performing pairing and bonding processes on the closed surface set to obtain bonded tight surfaces include: The surface patches in the closed surface set are cut twice, and the cutting condition is that the surface patches connect at least three spatially non-adjacent external regions. Select one of the surface patches as the initial surface, and locate a boundary curve of the initial surface; Find another surface patch that shares the boundary curve and is located on the other side of the boundary curve, and pair it with the initial surface to form the external boundary of the local three-dimensional solid. Two surface patches are topologically bonded along the boundary curve to form a new surface combination. The new surface combination is then used as the processing object to pair and bond the remaining boundary with other surface patches until the formed surface meets the requirements of tightly bonded surfaces. Finally, the surface patches that participated in the construction of the surface are removed from the closed surface set. The remaining surface patches continue to repeat the pairing and bonding process until the closed surface set no longer has surface patches available for pairing.

9. The method for representing the boundary of a three-dimensional solid based on a partially ordered set of bonded tight surfaces according to claim 4, characterized in that, The partial order relation defines inclusion and coverage relations; wherein, the inclusion relation is defined as if and only if the surfaces in the final result set... The bounded region enclosed by the surface is completely contained within the surface. Within the bounded region enclosed by the boundary, the covering relationship is defined as if And there is no other surface. Make Then it is called cover This indicates a direct inclusion relationship.

10. The method for representing the boundary of a three-dimensional solid based on a partially ordered set of bonded tight surfaces according to claim 1, characterized in that, The topological invariants include the number of connected components and the number of genus; the number of connected components is equal to the number of positively oriented, tightly bound surfaces, and the number of genus is derived within the time complexity based on the number, type, and coverage relationship of nodes in each branch of the partial order set using Euler's characteristic formula.

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