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We present a new algorithm for conformal mesh parameterization. It is based on a precise notion of <i>discrete conformal equivalence</i> for triangle meshes which mimics the notion of conformal equivalence for smooth surfaces. The problem of finding a flat mesh that is discretely conformally equivalent to a given mesh can be solved efficiently by minimizing… (More)

We define a discrete Laplace-Beltrami operator for simplicial surfaces (Definition 16). It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finite-elements Laplacian (the so called " cotan formula ") except that it is based on the intrinsic Delaunay triangula-tion of… (More)

We introduce a novel method for the construction of discrete conformal mappings from surface meshes of arbitrary topology to the plane. Our approach is based on <i>circle patterns</i>, that is, 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… (More)

We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay trian-gulations may be interpreted as images of hyperbolic polyhedra with one vertex on and the remaining… (More)

We prove existence and uniqueness results for patterns of circles with prescribed intersection angles on constant curvature surfaces. Our method is based on two new functionals—one for the Euclidean and one for the hyperbolic case. We show how Colin deVerdì ere's, Brägger's and Rivin's functionals can be derived from ours.

A Delaunay cell decomposition of a surface with constant curvature gives rise to a circle pattern, consisting of the circles which are circumscribed to the facets. We treat the problem whether there exists a Delaunay cell decomposition for a given (topological) cell decomposition and given intersection angles of the circles, whether it is unique and how it… (More)

- Alexander I. Bobenko, Tatyana V.Pavlyukevich, Boris A. Springborn
- 2002

For minimal surfaces in R 3 there is a representation, due to Weierstrass, in terms of holomorphic data. The Gauss-Codazzi equations for minimal surfaces in R 3 are equivalent to those for surfaces in hyperbolic space with constant mean curvature 1 (CMC-1 surfaces). This lead Bryant [Br] to derive a representation for CMC-1 surfaces in terms of holomorphic… (More)

The discrete Laplace-Beltrami operator plays a prominent role in many Digital Geometry Processing applications ranging from denoising to parameterization, editing, and physical simulation. The standard discretization uses the cotangents of the angles in the immersed mesh which leads to a variety of numerical problems. We advocate use of the intrinsic… (More)

It is known that for each combinatorial type of convex 3-dimensional polyhedra, there is a representative with edges tangent to the unit sphere. This representative is unique up to projective transformations that fix the unit sphere. We show that there is a unique representative (up to congruence) with edges tangent to the unit sphere such that the origin… (More)