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A05
Conformal Deformations of Discrete Surfaces

The Construction of a Discrete Differential Geometry Version of Certain Conformal Deformations

Two geometries can be considered equivalent if there exists an angle preserving transformation between them; this is a so called conformal transformation. In the smooth case, conformal equivalences are quite well understood. However, mimicking their construction in the discrete case brought up not only interesting properties and algorithms, but also interesting problems - first and foremost the question of how to construct conformal deformations with certain prescribed properties.

Scientific Details+

The goal of this project is to develop an exact Discrete Differential Geometry version of conformal deformations with prescribed mean curvature half-density. This will be based on the notion of conformal equivalence of triangle meshes introduced in B. Springborn, P. Schröder, and U. Pinkall. Conformal equivalence of triangle meshes , and provide an exact Discrete Differential Geometry version of the numerical methods developed in K. Crane, U. Pinkall, and P. Schröder. Spin transformations of discrete surfaces.

Publications+

Papers
Holomorphic vector fields and quadratic differentials on planar triangular meshes

Authors: Lam, W. Y. and Pinkall, U.
In Collection: Advances in Discrete Differential Geometry, Springer
Date: 2016
Download: arXiv

Isothermic Triangulated Surfaces

Authors: Lam, Wai Yeung and Pinkall, Ulrich
Note: preprint
Date: Jan 2015
Download: arXiv

Smoke Rings from Smoke

Authors: Weißmann, Steffen and Pinkall, Ulrich and Schröder, Peter
Journal: ACM Trans. Graph., 33(4):140:1--140:8
Date: Jul 2014
DOI: 10.1145/2601097.2601171
Download: external preprint

Globally optimal direction fields

Authors: Knöppel, Felix and Crane, Keenan and Pinkall, Ulrich and Schröder, Peter
Journal: ACM Trans. Graph., 32(4)
Date: 2013
Download: external

Robust Fairing via Conformal Curvature Flow

Authors: Crane, Keenan and Pinkall, Ulrich and Schröder, Peter
Journal: ACM Trans. Graph., 32
Date: 2013
Download: external

Spin transformations of discrete surfaces

Authors: Crane, K. and Pinkall, U. and Schröder, P.
Journal: ACM Transactions on Graphics, 30
Date: 2011
Download: external

A simple geometric model for elastic deformations

Authors: Chao, I. and Pinkall, U. and Sanan, P. and Schröder, P.
Journal: ACM Transactions on Graphics, 29
Date: 2010
Download: external

Conformal equivalence of triangle meshes

Authors: Springborn, B. and Schröder, P. and Pinkall, U.
Journal: ACM Transactions on Graphics, 27
Date: 2008
DOI: 10.1145/1360612.1360676
Download: external

Quaternionic holomorphic geometry: Plücker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori

Authors: Ferus, D. and Leschke, K. and Pedit, F. and Pinkall, U.
Journal: Inventiones Mathematicae, 146(3):507--593
Date: 2001
Download: arXiv

Discretization of surfaces and integrable systems

Authors: Bobenko, A. and Pinkall, U.
In Collection: Discrete integrable geometry and physics, Oxford Univ. Press
Date: 1999
Download: external

Computing discrete minimal surfaces and their conjugates

Authors: Pinkall, U. and Polthier, K.
Journal: Experimental Mathematics, 2(1):15--36
Date: 1993
Download: external

Bonnet pairs and isothermic surfaces

Authors: Kamberov, G. and Pedit, F. and Pinkall, U.
Journal: Duke Math. J., 92(3):637--644
Date:
DOI: 10.1215/S0012-7094-98-09219-5
Download: external arXiv


Master thesis
Infinitesimal conformal deformations of triangulated surfaces

Author: Lam, W.Y.
Journal: TU Berlin
Date: 2013
Download: internal


Team+

Prof. Dr. Ulrich Pinkall   +

Projects: A05, C07
University: TU Berlin
E-Mail: pinkall[at]math.tu-berlin.de
Website: http://page.math.tu-berlin.de/~pinkall/

Dr. Albert Chern   +

Projects: A05, C07
University: TU Berlin
E-Mail: chern[at]math.tu-berlin.de