Premium
Applied Geometry: Discrete Differential Calculus for Graphics
Author(s) -
Desbrun Mathieu
Publication year - 2004
Publication title -
computer graphics forum
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.578
H-Index - 120
eISSN - 1467-8659
pISSN - 0167-7055
DOI - 10.1111/j.1467-8659.2004.00756.x
Subject(s) - discrete geometry , discretization , geometry , surface (topology) , differential geometry , digital geometry , computational geometry , parametric surface , smoothing , graphics , differential (mechanical device) , mathematics , computer science , computer graphics (images) , mathematical analysis , physics , artificial intelligence , parametric statistics , digital image , statistics , image (mathematics) , thermodynamics , image processing
Geometry has been extensively studied for centuries, almost exclusively from a differential point of view. However, with the advent of the digital age, the interest directed to smooth surfaces has now partially shifted due to the growing importance of discrete geometry. From 3D surfaces in graphics to higher dimensional manifolds in mechanics, computational sciences must deal with sampled geometric data on a daily basis‐hence our interest in Applied Geometry.In this talk we cover different aspects of Applied Geometry. First, we discuss the problem of Shape Approximation, where an initial surface is accurately discretized (i.e., remeshed) using anisotropic elements through error minimization. Second, once we have a discrete geometry to work with, we briefly show how to develop a full‐ blown discrete calculus on such discrete manifolds, allowing us to manipulate functions, vector fields, or even tensors while preserving the fundamental structures and invariants of the differential case. We will emphasize the applicability of our discrete variational approach to geometry by showing results on surface parameterization, smoothing, and remeshing, as well as virtual actors and thin‐shell simulation.Joint work with: Pierre Alliez (INRIA), David Cohen‐Steiner (Duke U.), Eitan Grinspun (NYU), Anil Hirani (Caltech), Jerrold E. Marsden (Caltech), Mark Meyer (Pixar), Fred Pighin (USC), Peter Schröder (Caltech), Yiying Tong (USC).