Smoothness of Subdivision Surfaces with Boundary

Henning Biermann; Sara Grundel; Denis Zorin

doi:10.1007/s00365-015-9292-4

Smoothness of Subdivision Surfaces with Boundary

Henning Biermann, Sara Grundel, Denis Zorin

Computer Science

Research output: Contribution to journal › Article › peer-review

Abstract

Subdivision rules for meshes with boundary are essential for practical applications of subdivision surfaces. These rules have to result in piecewise C^ℓ-continuous boundary limit curves and ensure C^ℓ-continuity of the surface itself. Extending the theory of Zorin (Constr Approx 16(3):359–397, 2000), we present in this paper general necessary and sufficient conditions for C^ℓ-continuity of subdivision schemes for surfaces with boundary, and specialize these to practically applicable sufficient conditions for ^C1-continuity. We use these conditions to show that certain boundary rules for Loop and Catmull–Clark are in fact C¹ continuous.

Original language	English (US)
Pages (from-to)	1-29
Number of pages	29
Journal	Constructive Approximation
Volume	42
Issue number	1
DOIs	https://doi.org/10.1007/s00365-015-9292-4
State	Published - Aug 1 2015

Keywords

Boundary rules
C-analysis
Catmull–Clark
Characteristic maps
Loop
Subdivision algorithms

ASJC Scopus subject areas

Analysis
General Mathematics
Computational Mathematics

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10.1007/s00365-015-9292-4

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AU - Biermann, Henning

AU - Grundel, Sara

AU - Zorin, Denis

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AB - Subdivision rules for meshes with boundary are essential for practical applications of subdivision surfaces. These rules have to result in piecewise Cℓ-continuous boundary limit curves and ensure Cℓ-continuity of the surface itself. Extending the theory of Zorin (Constr Approx 16(3):359–397, 2000), we present in this paper general necessary and sufficient conditions for Cℓ-continuity of subdivision schemes for surfaces with boundary, and specialize these to practically applicable sufficient conditions for C1-continuity. We use these conditions to show that certain boundary rules for Loop and Catmull–Clark are in fact C1 continuous.

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KW - C-analysis

KW - Catmull–Clark

KW - Characteristic maps

KW - Loop

KW - Subdivision algorithms

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Smoothness of Subdivision Surfaces with Boundary

Abstract

Keywords

ASJC Scopus subject areas

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