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Computer Graphics - TU Berlin
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Abstract

While the theory and applications of discrete Laplacians on triangulated surfaces are abundantly developed, far less is known about the general polygonal case. We present here a principled approach for constructing geometric discrete Laplacians on surfaces with arbitrary polygonal faces, encompassing non-planar and non-convex polygons. Our construction is guided by closely mimicking structural properties of the smooth Laplace-Beltrami operator. Among other features, our construction leads to a methodical extension of the widely employed cotan formula from triangles to polygons. Besides carefully laying out theoretical aspects, we demonstrate the versatility of our approach for a variety of geometry processing applications, embarking on situations that would have been more difficult to achieve based on geometric Laplacians for simplicial meshes or purely combinatorial Laplacians for general meshes.

Documents

Code

Compilation

In order to build the source code as well as the required libraries (OpenMesh, Eigen) the CMake build system is necessary. The Eigen library is "header-only", i.e. no compilation is required, but CMake provides a convenient way for system-wide installation. Package management systems (MacPorts, APT) provide an easy way to install many of the required packages. Unfortunately, the availability of packages is not consistent over all systems.

Compilation on Mac OS X

install MacPorts (if not installed yet)
install available packages ... $ sudo port install cmake eigen3 qt4-mac SuiteSparse uncompress OpenMesh ... $ cd OpenMesh-2.0.1
$ mkdir build
$ cd build
$ cmake -DCMAKE_BUILD_TYPE=Release ..
$ make
$ sudo make install
 uncompress polymesh-src ... $ cd polymesh-src
$ mkdir build
$ cd build
$ cmake -DCMAKE_PREFIX_PATH="/opt/local" ..
$ make
 done.