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Conformal Geometry Processing


Crane, Keenan Michael (2013) Conformal Geometry Processing. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/8V9Z-N286.


This thesis introduces fundamental equations and numerical methods for manipulating surfaces in three dimensions via conformal transformations. Conformal transformations are valuable in applications because they naturally preserve the integrity of geometric data. To date, however, there has been no clearly stated and consistent theory of conformal transformations that can be used to develop general-purpose geometry processing algorithms: previous methods for computing conformal maps have been restricted to the flat two-dimensional plane, or other spaces of constant curvature. In contrast, our formulation can be used to produce---for the first time---general surface deformations that are perfectly conformal in the limit of refinement. It is for this reason that we commandeer the title Conformal Geometry Processing.

The main contribution of this thesis is analysis and discretization of a certain time-independent Dirac equation, which plays a central role in our theory. Given an immersed surface, we wish to construct new immersions that (i) induce a conformally equivalent metric and (ii) exhibit a prescribed change in extrinsic curvature. Curvature determines the potential in the Dirac equation; the solution of this equation determines the geometry of the new surface. We derive the precise conditions under which curvature is allowed to evolve, and develop efficient numerical algorithms for solving the Dirac equation on triangulated surfaces.

From a practical perspective, this theory has a variety of benefits: conformal maps are desirable in geometry processing because they do not exhibit shear, and therefore preserve textures as well as the quality of the mesh itself. Our discretization yields a sparse linear system that is simple to build and can be used to efficiently edit surfaces by manipulating curvature and boundary data, as demonstrated via several mesh processing applications. We also present a formulation of Willmore flow for triangulated surfaces that permits extraordinarily large time steps and apply this algorithm to surface fairing, geometric modeling, and construction of constant mean curvature (CMC) surfaces.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:signal processing; differential geometry
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Computer Science
Awards:Everhart Distinguished Graduate Student Lecturer Award, 2012.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Schroeder, Peter (advisor)
  • Desbrun, Mathieu (co-advisor)
Thesis Committee:
  • Schroeder, Peter (chair)
  • Desbrun, Mathieu
  • Markovic, Vladimir
  • Pinkall, Ulrich
Defense Date:15 April 2013
Funding AgencyGrant Number
GooglePhD Fellowship
Hausdorff Research Institute for MathematicsUNSPECIFIED
BMBF Research Project GEOMECSFB / Transregio 109 "Discretization in Geometry and Dynamics"
TU München Institute for Advanced StudyGerman Excellence Initiative
DFG Research Center MatheonUNSPECIFIED
DFG Research Unit Polyhedral SurfacesUNSPECIFIED
Record Number:CaltechTHESIS:06052013-020706629
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:7837
Deposited By: Keenan Crane
Deposited On:06 Jun 2013 23:47
Last Modified:04 Oct 2019 00:01

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