Parametric Surface Creation at Extraordinary Vertices
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Solution Overview
Problem
Existing CAD systems face challenges in creating parametric surfaces with sufficient geometrical continuity, especially at extraordinary vertices, which leads to curvature discontinuities and difficulties in producing aesthetically smooth shapes and offset surfaces.
Innovation Solution
A computer-implemented method that uses sweeping and loft algorithms to define parametric elementary surfaces, matches them with the base mesh, and trims surfaces to achieve required geometrical continuity, including G1 or G2 continuity, while being compatible with sharp edges, by subdividing the base mesh according to Catmull-Clark rules and solving systems of equations to enforce continuity.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Manufacturing precision
If conventional parametric surface creation methods are used, then the modeling process is simple, but curvature continuity is insufficient at extraordinary vertices
Solution Approach 1:
The base mesh is divided into regular and extraordinary vertices/patches. Regular patches use conventional parametric representation while extraordinary patches are segmented out and handled separately with specialized algorithms to ensure curvature continuity.
Solution Approach 2:
Different mathematical representations are applied to different regions: conventional NURBS patches for regular areas and specially constructed patches with enforced curvature continuity for extraordinary vertices. This local differentiation resolves the continuity problem without complicating the entire modeling process.
2Shape
If subdivision surfaces are used to create smooth shapes, then surface smoothness is improved, but compatibility with sharp edges is lost
Solution Approach 1:
The method applies different continuity requirements to different regions: G2 curvature continuity is enforced for smooth surfaces while allowing G0 discontinuities at sharp edges. This local differentiation enables both smooth shapes and sharp edges to coexist in the same model.
Solution Approach 2:
The continuity level is made dynamic and adaptable based on the local geometry requirements. The algorithm automatically adjusts the continuity enforcement level depending on whether the region requires smoothness or can accommodate sharp edges, providing versatility in handling different geometric features.
3Manufacturing precision
If extraordinary vertices are handled with conventional patches, then the modeling process is straightforward, but curvature discontinuities occur
Solution Approach 1:
An intermediate system of equations is introduced to mediate between the simple patch construction and the required curvature continuity. The equations serve as a bridge that enforces G2 continuity constraints while maintaining relatively straightforward patch construction procedures.
Solution Approach 2:
The method modifies the mathematical parameters of the patches near extraordinary vertices to ensure curvature continuity. By adjusting control points and surface parameters locally, the invention achieves high curvature precision without fundamentally changing the overall patch construction approach.
4Shape
If G2 curvature continuity is enforced everywhere, then surface smoothness is maximized, but offset surface creation becomes problematic at extraordinary vertices
Solution Approach 1:
The method applies G2 curvature continuity locally at extraordinary vertices rather than uniformly everywhere. This selective enforcement maintains surface smoothness where needed while avoiding the mathematical singularities that prevent offset surface creation, thereby enabling offset operations at extraordinary vertices.
Data Source
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AI summary
The invention is directed to a computer program product, a computer system and a computer-implemented method for creating a parametric surface (16), comprising the steps of : - providing a base mesh (15) having a plurality of outer or inner vertices (24, 25, 101) with valence greater than two, said vertices defining faces and at least one of said vertices being an extraordinary vertex (25, 101) defined as being different from: - an outer vertex with valence two; - an outer vertex with valence three; or - a inner vertex with valence four (24), - converting faces of the base mesh into parametric elementary surfaces (103, 104, 105); - trimming at least one (105) of said parametric elementary surfaces located in the vicinity of said extraordinary vertex (101), according to: - at least two parametric elementary surfaces (103, 104) adjacent to said at least one parametric elementary surface (111); and - the base mesh (15); and - constructing at least one parametric elementary surface (123, 124), adjacent to said trimmed parametric elementary surface (102), wherein the resulting parametric elementary surfaces (102, 103, 104, 123, 124) form the parametric surface (16).