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Application Area
- F (48) (remove)
We define a discrete Laplace-Beltrami operator for simplicial surfaces. It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the one defined by Pinkall and Polthier (the so called “cotan formula”) except that it is based on the intrinsic Delaunay triangulation of the simplicial surface. This leads to new definitions of discrete harmonic and holomorphic functions, discrete mean curvature, and discrete minimal surfaces.
We prove an existence and uniqueness theorem for weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations can also be interpreted as hyperbolic polyhedra with vertices beyond the infinite boundary. The proof is based on a variational principle. This extends similar work by Rivin on Delaunay triangulations and ideal polyhedra to weighted Delaunay triangulations and hyperideal polyhedra.
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 circle patterns, i.e., 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. We demonstrate the versatility and performance of our algorithm with a variety of examples.
We introduce FreeLence, a lossless single-rate connectivity compression algorithm for triangle surface meshes. Based upon a geometry-driven traversal scheme we present two novel and simple concepts: free-valence connectivity encoding and entropy coding based on geometric context. Together these techniques yield signicantly smaller rates for connectivity compression than current state of the art approaches - valence-based algorithms and Angle- Analyzer, with an average of 36% improvement over the former and an average of 18% over the latter on benchmark 3D models, combined with the ability to well adapt to the regularity of meshes. We also prove that our algorithm exhibits a smaller worst case entropy
for a class of ”well-behaved” triangle meshes than valence-driven connectivity encoding approaches.
We provide conditions for convergence of polyhedral surfaces and their
discrete geometric properties to smooth surfaces embedded in R^3. The
notion of totally normal convergence is shown to be equivalent to the convergence
of either one of the following: surface area, intrinsic metric, and
Laplace-Beltrami operators. We further show that totally normal convergence
implies convergence results for shortest geodesics, mean curvature,
and solutions to the Dirichlet problem. This work provides the justifi-
cation for a discrete theory of differential geometric operators defined on
polyhedral surfaces based on a variational formulation.
A new method for noise removal of arbitrary surfaces
meshes is presented which focuses on the preservation
and sharpening of non-linear geometric features such
as curved surface regions and feature lines. Our method
uses a prescribed mean curvature flow (PMC) for simplicial
surfaces which is based on three new contributions:
1. the definition and efficient calculation of a
discrete shape operator and principal curvature properties
on simplicial surfaces that is fully consistent with
the well-known discrete mean curvature formula, 2. an
anisotropic discrete mean curvature vector that combines
the advantages of the mean curvature normal with
the special anisotropic behaviour along feature lines of
a surface, and 3. an anisotropic prescribed mean curvature
flow which converges to surfaces with an estimated
mean curvature distribution and with preserved nonlinear
features. Additionally, the PMC flow prevents
boundary shrinkage at constrained and free boundary
segments.
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 is the barycenter of the
points where the edges touch the sphere.