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A sharp degree bound on G 2-refinable multi-sided surfaces.

Kȩstutis Karčiauskas1, Jörg Peters2

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|June 17, 2020
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Summary
This summary is machine-generated.

Achieving greater flexibility in multi-sided surfaces requires a higher polynomial degree. This study identifies the minimum degree needed for refinable surfaces, demonstrating a bi-6 construction that enhances modeling and engineering analysis.

Keywords:
flexibility-increasingfree-form surfacegeneralized splinegood shapelower boundmulti-sidedrefinement

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Area of Science:

  • Computer-Aided Design (CAD)
  • Geometric Modeling
  • Spline Theory

Background:

  • Spline-based surfaces are crucial for design and analysis.
  • Current multi-sided G2 continuous surfaces lack sufficient flexibility when using low polynomial degrees.
  • Refinement of spline spaces should ideally increase degrees of freedom.

Purpose of the Study:

  • To establish a tight lower bound on the polynomial degree for refinable multi-sided G2 spline surfaces.
  • To demonstrate that lower degrees (e.g., bi-5) are insufficient for flexibility.
  • To present a viable construction for enhanced surface modeling.

Main Methods:

  • Analysis of C2 spline complexes for multi-sided surface constructions.
  • Investigating the relationship between polynomial degree and surface flexibility.
  • Developing and evaluating a multi-sided bi-6 construction using 2x2 macro-patches.

Main Results:

  • A tight lower bound on the polynomial degree for flexibility-increasing refinable multi-sided G2 surfaces is established.
  • Bi-5 constructions are shown to be insufficient for achieving desired flexibility.
  • A multi-sided bi-6 construction demonstrates good highlight line and curvature distributions.

Conclusions:

  • Higher polynomial degrees are essential for achieving flexibility in multi-sided G2 spline surfaces.
  • The proposed bi-6 construction offers a practical solution for improved surface modeling and engineering analysis.
  • This work advances the understanding of spline surface refinement and its impact on design capabilities.