Computing extremal quasiconformal maps

Ofir Weber, Ashish Myles, Denis Zorin

Research output: Contribution to journalArticle

Abstract

Conformal maps are widely used in geometry processing applications. They are smooth, preserve angles, and are locally injective by construction. However, conformal maps do not allow for boundary positions to be prescribed. A natural extension to the space of conformal maps is the richer space of quasiconformal maps of bounded conformal distortion. Extremal quasiconformal maps, that is, maps minimizing the maximal conformal distortion, have a number of appealing properties making them a suitable candidate for geometry processing tasks. Similarly to conformal maps, they are guaranteed to be locally bijective; unlike conformal maps however, extremal quasiconformal maps have sufficient flexibility to allow for solution of boundary value problems. Moreover, in practically relevant cases, these solutions are guaranteed to exist, are unique and have an explicit characterization. We present an algorithm for computing piecewise linear approximations of extremal quasiconformal maps for genus-zero surfaces with boundaries, based on Teichmüller's characterization of the dilatation of extremal maps using holomorphic quadratic differentials.We demonstrate that the algorithm closely approximates the maps when an explicit solution is available and exhibits good convergence properties for a variety of boundary conditions.

Original languageEnglish (US)
Pages (from-to)1679-1689
Number of pages11
JournalComputer Graphics Forum
Volume31
Issue number5
DOIs
StatePublished - 2012

Fingerprint

Geometry
Processing
Boundary value problems
Boundary conditions

Keywords

  • Conformal parameterization
  • Parameterization
  • Quadrangulation
  • Remeshing

ASJC Scopus subject areas

  • Computer Networks and Communications

Cite this

Computing extremal quasiconformal maps. / Weber, Ofir; Myles, Ashish; Zorin, Denis.

In: Computer Graphics Forum, Vol. 31, No. 5, 2012, p. 1679-1689.

Research output: Contribution to journalArticle

Weber, Ofir ; Myles, Ashish ; Zorin, Denis. / Computing extremal quasiconformal maps. In: Computer Graphics Forum. 2012 ; Vol. 31, No. 5. pp. 1679-1689.
@article{1e3e1b6cc5e449288d49964413b633ae,
title = "Computing extremal quasiconformal maps",
abstract = "Conformal maps are widely used in geometry processing applications. They are smooth, preserve angles, and are locally injective by construction. However, conformal maps do not allow for boundary positions to be prescribed. A natural extension to the space of conformal maps is the richer space of quasiconformal maps of bounded conformal distortion. Extremal quasiconformal maps, that is, maps minimizing the maximal conformal distortion, have a number of appealing properties making them a suitable candidate for geometry processing tasks. Similarly to conformal maps, they are guaranteed to be locally bijective; unlike conformal maps however, extremal quasiconformal maps have sufficient flexibility to allow for solution of boundary value problems. Moreover, in practically relevant cases, these solutions are guaranteed to exist, are unique and have an explicit characterization. We present an algorithm for computing piecewise linear approximations of extremal quasiconformal maps for genus-zero surfaces with boundaries, based on Teichm{\"u}ller's characterization of the dilatation of extremal maps using holomorphic quadratic differentials.We demonstrate that the algorithm closely approximates the maps when an explicit solution is available and exhibits good convergence properties for a variety of boundary conditions.",
keywords = "Conformal parameterization, Parameterization, Quadrangulation, Remeshing",
author = "Ofir Weber and Ashish Myles and Denis Zorin",
year = "2012",
doi = "10.1111/j.1467-8659.2012.03173.x",
language = "English (US)",
volume = "31",
pages = "1679--1689",
journal = "Computer Graphics Forum",
issn = "0167-7055",
publisher = "Wiley-Blackwell",
number = "5",

}

TY - JOUR

T1 - Computing extremal quasiconformal maps

AU - Weber, Ofir

AU - Myles, Ashish

AU - Zorin, Denis

PY - 2012

Y1 - 2012

N2 - Conformal maps are widely used in geometry processing applications. They are smooth, preserve angles, and are locally injective by construction. However, conformal maps do not allow for boundary positions to be prescribed. A natural extension to the space of conformal maps is the richer space of quasiconformal maps of bounded conformal distortion. Extremal quasiconformal maps, that is, maps minimizing the maximal conformal distortion, have a number of appealing properties making them a suitable candidate for geometry processing tasks. Similarly to conformal maps, they are guaranteed to be locally bijective; unlike conformal maps however, extremal quasiconformal maps have sufficient flexibility to allow for solution of boundary value problems. Moreover, in practically relevant cases, these solutions are guaranteed to exist, are unique and have an explicit characterization. We present an algorithm for computing piecewise linear approximations of extremal quasiconformal maps for genus-zero surfaces with boundaries, based on Teichmüller's characterization of the dilatation of extremal maps using holomorphic quadratic differentials.We demonstrate that the algorithm closely approximates the maps when an explicit solution is available and exhibits good convergence properties for a variety of boundary conditions.

AB - Conformal maps are widely used in geometry processing applications. They are smooth, preserve angles, and are locally injective by construction. However, conformal maps do not allow for boundary positions to be prescribed. A natural extension to the space of conformal maps is the richer space of quasiconformal maps of bounded conformal distortion. Extremal quasiconformal maps, that is, maps minimizing the maximal conformal distortion, have a number of appealing properties making them a suitable candidate for geometry processing tasks. Similarly to conformal maps, they are guaranteed to be locally bijective; unlike conformal maps however, extremal quasiconformal maps have sufficient flexibility to allow for solution of boundary value problems. Moreover, in practically relevant cases, these solutions are guaranteed to exist, are unique and have an explicit characterization. We present an algorithm for computing piecewise linear approximations of extremal quasiconformal maps for genus-zero surfaces with boundaries, based on Teichmüller's characterization of the dilatation of extremal maps using holomorphic quadratic differentials.We demonstrate that the algorithm closely approximates the maps when an explicit solution is available and exhibits good convergence properties for a variety of boundary conditions.

KW - Conformal parameterization

KW - Parameterization

KW - Quadrangulation

KW - Remeshing

UR - http://www.scopus.com/inward/record.url?scp=84879362903&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84879362903&partnerID=8YFLogxK

U2 - 10.1111/j.1467-8659.2012.03173.x

DO - 10.1111/j.1467-8659.2012.03173.x

M3 - Article

VL - 31

SP - 1679

EP - 1689

JO - Computer Graphics Forum

JF - Computer Graphics Forum

SN - 0167-7055

IS - 5

ER -