Canonical Möbius subdivision

Amir Vaxman; Christian Müller; Ofir Weber

doi:10.1145/3272127.3275007

Canonical Möbius subdivision

Amir Vaxman, Christian Müller, Ofir Weber

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

Abstract

We present a novel framework for creating Möbius-invariant subdivision operators with a simple conversion of existing linear subdivision operators. By doing so, we create a wide variety of subdivision surfaces that have properties derived from Möbius geometry; namely, reproducing spheres, circular arcs, and Möbius regularity. Our method is based on establishing a canonical form for each 1-ring in the mesh, representing the class of all 1-rings that are Möbius equivalent to that 1-ring. We perform a chosen linear subdivision operation on these canonical forms, and blend the positions contributed from adjacent 1-rings, using two novel Möbius-invariant operators, into new face and edge points. The generality of the method allows for easy coarse-to-fine mesh editing with diverse polygonal patterns, and with exact reproduction of circular and spherical features. Our operators are in closed-form and their computation is as local as the computation of the linear operators they correspond to, allowing for efficient subdivision mesh editing and optimization.

Original language	English
Title of host publication	SIGGRAPH Asia 2018 Technical Papers, SIGGRAPH Asia 2018
Publisher	Association for Computing Machinery
ISBN (Electronic)	9781450360081
DOIs	https://doi.org/10.1145/3272127.3275007
Publication status	Published - 4 Dec 2018
Event	SIGGRAPH Asia 2018 Technical Papers - International Conference on Computer Graphics and Interactive Techniques, SIGGRAPH Asia 2018 - Tokyo, Japan Duration: 4 Dec 2018 → 7 Dec 2018

Conference

Conference	SIGGRAPH Asia 2018 Technical Papers - International Conference on Computer Graphics and Interactive Techniques, SIGGRAPH Asia 2018
Country/Territory	Japan
City	Tokyo
Period	4/12/18 → 7/12/18

Funding

This research was partially funded by the Austrian Science Fund (FWF) (projects P 29981 and I 2978-N35) and by the Israel Science Foundation (grants No. 1869/15 and 2102/15). We thank Ron Van-derfeesten for the stylistic rendering of Figure 2.

Keywords

Architectural geometry
Conformal transformations
Mesh subdivision
Möbius transformations
Regular meshes

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10.1145/3272127.3275007

a227-vaxmanFinal published version, 78.7 MBLicence: CC BY-NC

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title = "Canonical M{\"o}bius subdivision",

abstract = "We present a novel framework for creating M{\"o}bius-invariant subdivision operators with a simple conversion of existing linear subdivision operators. By doing so, we create a wide variety of subdivision surfaces that have properties derived from M{\"o}bius geometry; namely, reproducing spheres, circular arcs, and M{\"o}bius regularity. Our method is based on establishing a canonical form for each 1-ring in the mesh, representing the class of all 1-rings that are M{\"o}bius equivalent to that 1-ring. We perform a chosen linear subdivision operation on these canonical forms, and blend the positions contributed from adjacent 1-rings, using two novel M{\"o}bius-invariant operators, into new face and edge points. The generality of the method allows for easy coarse-to-fine mesh editing with diverse polygonal patterns, and with exact reproduction of circular and spherical features. Our operators are in closed-form and their computation is as local as the computation of the linear operators they correspond to, allowing for efficient subdivision mesh editing and optimization.",

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author = "Amir Vaxman and Christian M{\"u}ller and Ofir Weber",

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AU - Müller, Christian

AU - Weber, Ofir

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N2 - We present a novel framework for creating Möbius-invariant subdivision operators with a simple conversion of existing linear subdivision operators. By doing so, we create a wide variety of subdivision surfaces that have properties derived from Möbius geometry; namely, reproducing spheres, circular arcs, and Möbius regularity. Our method is based on establishing a canonical form for each 1-ring in the mesh, representing the class of all 1-rings that are Möbius equivalent to that 1-ring. We perform a chosen linear subdivision operation on these canonical forms, and blend the positions contributed from adjacent 1-rings, using two novel Möbius-invariant operators, into new face and edge points. The generality of the method allows for easy coarse-to-fine mesh editing with diverse polygonal patterns, and with exact reproduction of circular and spherical features. Our operators are in closed-form and their computation is as local as the computation of the linear operators they correspond to, allowing for efficient subdivision mesh editing and optimization.

AB - We present a novel framework for creating Möbius-invariant subdivision operators with a simple conversion of existing linear subdivision operators. By doing so, we create a wide variety of subdivision surfaces that have properties derived from Möbius geometry; namely, reproducing spheres, circular arcs, and Möbius regularity. Our method is based on establishing a canonical form for each 1-ring in the mesh, representing the class of all 1-rings that are Möbius equivalent to that 1-ring. We perform a chosen linear subdivision operation on these canonical forms, and blend the positions contributed from adjacent 1-rings, using two novel Möbius-invariant operators, into new face and edge points. The generality of the method allows for easy coarse-to-fine mesh editing with diverse polygonal patterns, and with exact reproduction of circular and spherical features. Our operators are in closed-form and their computation is as local as the computation of the linear operators they correspond to, allowing for efficient subdivision mesh editing and optimization.

KW - Architectural geometry

KW - Conformal transformations

KW - Mesh subdivision

KW - Möbius transformations

KW - Regular meshes

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M3 - Conference contribution

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BT - SIGGRAPH Asia 2018 Technical Papers, SIGGRAPH Asia 2018

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Y2 - 4 December 2018 through 7 December 2018

ER -

Canonical Möbius subdivision

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