Eigen Polyhedron Mesh Refinement for G1 Tangent Continuity
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Solution Overview
Problem
Conventional methods for refining complex surface models, such as Catmull-Clark and NURBS surfaces, often result in surface tangent discontinuities at extraordinary vertices, leading to artifacts like cusps and distortions in the represented object.
Innovation Solution
The method involves generating refinement rules based on eigen polyhedra in a plane, which are then applied to a non-uniform mesh in three-dimensional space, ensuring continuous tangent planes at extraordinary points by scaling and translating the eigen polyhedron, thus avoiding conventional refinement rules that cause discontinuities.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Manufacturing precision
If conventional refinement rules are applied to control points of a mesh, then the mesh can be refined from coarse to fine, but surface tangent discontinuities occur at extraordinary vertices causing artifacts like cusps and distortions
Solution Approach 1:
The patent changes the refinement parameters by introducing eigen polyhedron-based refinement rules that compute new control points using eigenvalues and eigenvectors of the refinement matrix. This mathematical transformation ensures that the refinement process maintains G1 continuity (continuous tangent planes) at extraordinary vertices, eliminating the artifacts caused by conventional refinement rules while achieving the desired fine mesh resolution
Solution Approach 2:
The patent introduces eigen polyhedra as an intermediary mathematical structure between the coarse mesh and the refined mesh. The eigen polyhedron serves as a mediator that transforms the refinement process, ensuring continuous tangent planes at extraordinary vertices by using its geometric properties and spectral characteristics to guide the refinement of control points
Data Source
AI summary
Techniques of refining a model of a complex surface can include deriving a set of refinement rules based on eigen polyhedra in a plane. Such rules reduce the refinement of a mesh in the plane to a scale and translation of the eigen polyhedron. Such refinement rules may then be applied to a non-uniform mesh in three-dimensional space having an extraordinary point in place of conventional refinement rules used in Catmull-Clark surfaces or NURBS surfaces. When these refinement rules are applied to a non-uniform mesh having an extraordinary point, the limiting surface is G1 at the extraordinary point, i.e., the tangent surfaces of the limiting mesh are continuous at the extraordinary point.