Polygon Triangulation Handling Degenerate Geometry
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Solution Overview
Problem
Existing polygon handling technologies face challenges in efficiently triangulating complex polygons with degeneracies, such as very short edges and small vertex angles, which can lead to inconsistent and erroneous geometric processing, especially in computer-aided manufacturing and computer animation applications.
Innovation Solution
A computer-implemented method that determines edge directions and vertex angles consistently, even for degenerate cases, by assigning reasonable directions to zero-length edges and using geometric properties to resolve ambiguous angles, allowing for robust triangulation of complex polygons with multiple boundaries and degenerate faces.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If conventional triangulation methods are used on complex polygons with degenerate edges and vertices, then processing speed may be maintained, but geometric consistency and computational reliability deteriorate due to ambiguous angle calculations
Solution Approach 1:
The patent applies preliminary action by pre-determining edge directions and vertex angles before performing triangulation. The method calculates and stores edge directions and signed angles in advance, creating a prepared geometric framework that guides subsequent triangulation operations. This preliminary preparation eliminates ambiguity during the actual triangulation process, ensuring geometric consistency without adding complexity to the core algorithm.
Solution Approach 2:
The patent introduces an intermediary computational framework that mediates between raw polygon data and triangulation operations. By inserting intermediate steps to determine edge directions and vertex angles with proper sign conventions, the method creates a bridge that translates ambiguous geometric data into consistent, reliable information suitable for triangulation, resolving the contradiction between reliability and complexity.
2Measurement precision
If standard angle calculation methods are applied to degenerate vertices, then computational simplicity is maintained, but measurement precision deteriorates due to ambiguous angle signs
Solution Approach 1:
The patent replaces traditional mechanical angle calculation methods with a sign-based computational system. Instead of relying on complex geometric constructions or iterative methods to resolve angle ambiguity, the invention substitutes a systematic approach using signed angles determined by edge directions and vertex winding order. This substitution achieves precise angle determination for degenerate vertices without the computational overhead of more complex methods.
Solution Approach 2:
The patent applies parameter changes by transforming the angle representation from unsigned to signed values. By changing the parameter representation to include sign information derived from edge directions and vertex winding, the method precisely captures angular information for degenerate vertices. This parameter transformation enables accurate angle determination while maintaining computational efficiency through straightforward sign assignment rules.
3Manufacturing precision
If degenerate polygons with zero-length edges are processed using traditional methods, then ease of operation is maintained, but manufacturing precision deteriorates due to inconsistent geometry
Solution Approach 1:
The patent applies local quality by treating degenerate edges and vertices with specialized handling rules distinct from normal geometry. The method identifies zero-length edges and assigns them directions based on adjacent edges, and treats degenerate vertices with specific angle calculation rules. This localized special treatment ensures manufacturing precision for problematic geometric features without complicating the processing of regular polygon elements.
Solution Approach 2:
The patent uses preliminary action to pre-process degenerate polygon features before triangulation. By预先 determining edge directions and vertex angles for degenerate cases, the method prepares a consistent geometric foundation that simplifies subsequent triangulation operations. This preliminary preparation eliminates the need for complex runtime checks and adjustments, maintaining ease of operation while ensuring triangulation accuracy.
Data Source
AI summary
A computer implemented method of handling polygons is disclosed, including polygons which are complex and include degeneracies. The method includes receiving an input polygon with at least one boundary, the at least one boundary having at least one vertex joining edges of the polygon boundary; determining an edge direction for each of the edges; determining a signed exterior or interior angle of each vertex angle wherein, if the exterior angle between the edges is −pi or pi to within a predetermined threshold, determining the signed angle of the vertex angle including assigning a sign based on the winding of the edges of the polygon boundary, a sequence of angles of polygon vertices which can be directly computed or retrieved from a memory, the geometrical property that all exterior angles of the polygon sum to −2pi or 2pi, and the known sign of the polygon area.