Parametric Surface Continuity at Singular Vertices
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Solution Overview
Problem
Existing CAD systems face challenges in creating parametric surfaces that meet specific geometrical continuity requirements, particularly at extraordinary vertices, and are not compatible with both open and closed vertices, leading to insufficient quality of continuity.
Innovation Solution
A process that provides a set of parametric elementary surfaces with internal continuity greater than or equal to the required geometrical continuity, and defines a system of equations to enforce continuity across edges joined to singular vertices, allowing for local and linear resolution, and considers both open and closed vertices.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Manufacturing precision
If conventional CAD systems use standard parametric surface modeling techniques, then the modeling process is simple and fast, but the geometrical continuity at extraordinary vertices is insufficient
Solution Approach 1:
The methodology segments the surface modeling problem by treating regular vertices and extraordinary vertices separately. Regular vertices use standard parametric surfaces with sufficient continuity, while extraordinary vertices are handled through a system of equations that enforces the required geometrical continuity. This segmentation allows the complex continuity requirement to be addressed locally at extraordinary vertices without complicating the entire modeling process.
Solution Approach 2:
The invention applies local quality by implementing different continuity enforcement strategies at different vertex types. At extraordinary vertices, a system of equations is solved to ensure the required geometrical continuity Gi, while at regular vertices, standard parametric surface continuity applies. This localized approach ensures high manufacturing precision at critical points without unnecessarily increasing overall process complexity.
2Manufacturing precision
If the system enforces strict geometrical continuity at all vertices, then surface quality improves, but the computational complexity and processing time increase significantly
Solution Approach 1:
The system segments vertices into regular and extraordinary categories, applying continuity enforcement only where necessary. Regular vertices inherently satisfy continuity requirements through standard parametric surface construction, while extraordinary vertices are the only locations requiring solution of the continuity system. This segmentation dramatically reduces computational overhead compared to enforcing continuity at all vertices.
Solution Approach 2:
The methodology applies partial action by enforcing the required geometrical continuity Gi only at extraordinary vertices rather than at all vertices. Since extraordinary vertices are the only locations where continuity cannot be guaranteed by standard parametric surfaces, this partial enforcement achieves the necessary surface quality without the excessive computational cost of universal continuity enforcement.
3Manufacturing precision
If conventional methods are used to handle extraordinary vertices, then the process is simpler, but the continuity between adjacent surfaces is insufficient
Solution Approach 1:
The system introduces an intermediary mathematical framework—a system of equations linking parameters of parametric surfaces across edges joined to extraordinary vertices. This intermediary structure mediates between the simple parametric surface definitions and the complex continuity requirements, enabling precise continuity enforcement at extraordinary vertices while maintaining relative ease of implementation through a systematic equation-solving approach.
Data Source
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AI summary
The invention is directed to a process for creating a parametric surface having a required geometrical continuity Gi, from a set of parametric elementary surfaces. Said process comprises a step of providing a set of parametric elementary surfaces, each elementary surface having edges, with vertices joining adjacent edges, each elementary surface having an internal continuity Cj at least equal to the required continuity Gi, the geometrical continuity between at least two elementary surfaces, across their common edge, being less than the required continuity Gi. Further, for each singular vertex, the process comprise steps of: defining a system of equations linking the parameters (including position coordinates of the usually called "control points") of the parametric elementary surfaces across all edges joined to the vertex and enforcing the required geometrical continuity across the joined edges; and solving the system of equations to obtain the parameters. Said singular vertex joins at least one edge across which the continuity between adjacent elementary surfaces is less than the required continuity Gi.