Good parameterizations are of central importance in many digital geometry processing tasks. Typically the behavior of such processing algorithms is related to the smoothness of the parameterization and how much distortion it contains. Since a parameterization maps a bounded region of the plane to the surface, a parameterization for a surface which is not homeomorphic to a disc must be made up of multiple pieces. We present a novel parameterization algorithm for arbitrary topology surface meshes which computes a globally smooth parameterization with low distortion. We optimize the patch layout subject to criteria such as shape quality and metric distortion, which are used to steer a mesh simplification approach for base complex construction. Global smoothness is achieved through simultaneous relaxation over all patches, with suitable transition functions between patches incorporated into the relaxation procedure. We demonstrate the quality of our parameterizations through numerical evaluation of distortion measures and the excellent rate distortion performance of semi-regular remeshes produced with these parameterizations. The numerical algorithms required to compute the parameterizations are robust and run on the order of minutes even for large meshes.

We introduce a new method for computing conformal transformations of triangle meshes in 3 . Conformal maps are desirable in digital geometry processing because they do not exhibit shear, and therefore preserve texture fidelity as well as the quality of the mesh itself. Traditional discretizations consider maps into the complex plane, which are useful only for problems such as surface parameterization and planar shape deformation where the target surface is flat. We instead consider maps into the quaternions , which allows us to work directly with surfaces sitting in 3 . In particular, we introduce a quaternionic Dirac operator and use it to develop a novel integrability condition on conformal deformations. Our discretization of this condition results in 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.

An algebraic commutative group G is associated to any vector field D on a complete algebraic variety X. The group G acts on X and its orbits are the minimal subvarieties of X which are tangent to D. This group is computed in the case of a vector field on P n .

Many computer graphics applications require high-intensity numerical simulation. We show that such computations can be performed efficiently on the GPU, which we regard as a full function streaming processor with high floating-point performance. We implemented two basic, broadly useful, computational kernels: a sparse matrix conjugate gradient solver and a regular-grid multigrid solver. Realtime applications ranging from mesh smoothing and parameterization to fluid solvers and solid mechanics can greatly benefit from these, evidence our example applications of geometric flow and fluid simulation running on NVIDIA's GeForce FX.

Figure 1: Progressively decoded isosurface (Headscan model) at 257 3 grid resolution (octree depth of eight). Up to the last octree level, no explicit geometry is encoded. Explicit geometry bits are only associated with the final resolution level. Distortions are relative L 2 errors measured by Metro [Cignoni et al. 1998], normalized to the bounding box diagonal length and given in multiples of 10 −4 .