3D Manifold Surface Generation via Delaunay Tetrahedralization
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Solution Overview
Problem
Conventional methods for generating a manifold surface from 3D curves often induce distortion and fail to provide a full surface representation due to projection into lower dimensions, leading to self-intersecting curves and limited types of 3D curves that can be used, restricting the complexity and accuracy of 3D modeling.
Innovation Solution
A system that generates a manifold surface by using Delaunay tetrahedralization directly in the 3D domain, connecting points of 3D curves in close spatial proximity, reducing the number of potential connections and surface patches to decrease computational expense and ensure non-intersecting surfaces, allowing for multiple manifold surface solutions based on metrics.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of manufacture
If conventional projection methods are used to parameterize curves into a 2D plane, then surface formation can be achieved through tessellation, but distortion is induced in the surface and self-intersecting curves are generated
Solution Approach 1:
The patent transitions from 2D plane-based tessellation to 3D space-based Delaunay tetrahedralization. Instead of projecting curves onto a 2D parameter domain and tessellating there, the method directly operates in 3D space by constructing a Delaunay tetrahedralization of the curve points and extracting surface triangles from the tetrahedron faces, thereby eliminating projection-induced distortion while maintaining computational feasibility.
2Reliability
If conventional 2D tessellation methods are used to connect projected curves, then a manifold surface can be generated, but the method fails when curves self-intersect or overlap in the parameter domain
Solution Approach 1:
By moving the tessellation operation from 2D parameter space to 3D object space, the patent eliminates the fundamental problem of self-intersection that plagues 2D methods. The Delaunay tetrahedralization in 3D naturally handles complex curve configurations without self-intersection issues, as the spatial relationships are preserved directly rather than being projected onto a 2D plane where overlaps occur.
Solution Approach 2:
The patent changes the fundamental parameters of the problem by working directly with 3D spatial coordinates rather than 2D parameterized coordinates. This parameter transformation allows the method to accept any 3D curve configuration (open, closed, intersecting, non-intersecting) without requiring the curves to be free of self-intersections in a parameter domain, thereby significantly increasing adaptability.
3Device complexity
If conventional methods restrict the types of 3D curves used for surface reconstruction, then computational complexity is reduced, but the complexity and accuracy of 3D modeling is limited
Solution Approach 1:
The patent changes the parameter domain from restricted 2D parameterized curves to unrestricted 3D spatial curves. By operating directly in 3D space with Delaunay tetrahedralization, the method can process any curve configuration without requiring curves to be parameterized in a specific way or to avoid self-intersections, thereby accepting more complex curve types while maintaining computational efficiency through the properties of Delaunay triangulation.
Data Source
AI summary
Various embodiments of a system and methods for generating a manifold surface of an object from a set of 3D curves which define the shape of the object are described. The set of 3D curves may be directly tessellated in the 3D domain to form a manifold surface with boundary. The tessellation may be a Delaunay tetrahedralization which represents the set of 3D curves. The surface of the tetrahedralization may contain a set of possible manifold surface solutions. A dual complex form of the tetrahedralization may be generated to reduce the number of possible manifold surface solutions. Surface patches may be generated from the dual complex form, dependent on a set of metrics that may further reduce the number of possible manifold surface solutions. Compatible surface patches may be combined to form a set of possible manifold surface solutions which may be displayed to a user.