A Caltech Library Service

Implementation of Circle Pattern Parameterization


Kharevych, Liliya (2006) Implementation of Circle Pattern Parameterization. Master's thesis, California Institute of Technology. doi:10.7907/PC38-GT79.


Circle Pattern is a novel method for the construction of discrete conformal mappings from surface meshes of arbitrary topology to the plane. This approach is based on representing mesh as arrangements of circles – one for each face – with prescribed intersection angles. Given these angles the circle radii follow as the unique minimizer of a convex energy. The method supports very flexible boundary conditions ranging from free boundaries to control of the boundary shape via prescribed curvatures. Closed meshes of genus zero can be parameterized over the sphere. To parameterize higher genus meshes we introduce cone singularities at designated vertices. The parameter domain is then a piecewise Euclidean surface. Cone singularities can also help to reduce the often very large area distortion of global conformal maps to moderate levels. Our method involves two optimization problems: a quadratic program and the unconstrained minimization of the circle pattern energy. The latter is a convex function of logarithmic radius variables with simple explicit expressions for gradient and Hessian. In this paper we demonstrate implementation details and possible extensions to the Circle Pattern method.

Item Type:Thesis (Master's thesis)
Subject Keywords:circle patterns; Conformal parameterizations; discrete analytic functions; discrete differential geometry; meshing; texture mapping
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Computer Science
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Schroeder, Peter
Thesis Committee:
  • Unknown, Unknown
Defense Date:22 May 2006
Record Number:CaltechETD:etd-05242006-224103
Persistent URL:
Related URLs:
URLURL TypeDescription DocumentSupplemental material
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:2016
Deposited By: Imported from ETD-db
Deposited On:05 Jun 2006
Last Modified:03 Oct 2019 23:13

Thesis Files

PDF (Thesis) - Final Version
See Usage Policy.

[img] HTML (CirclePatterns Documentation 1.0 ) - Supplemental Material
See Usage Policy.


Repository Staff Only: item control page