Applied Geometry: Discrete Differential Calculus for Graphics
Mathieu Desbrun
University of Southern California
Computer Science
Geometry has been extensively studied for centuries, almost exclusively from a differential point of view. However, with the advent of the digital age, interest in smooth surfaces has partially shifted due to the growing importance of discrete geometry. From three-dimensional 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 briefly 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 show how to develop a discrete differential 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 setting.
We emphasize the applicability of our discrete variational approach to geometry for medical applications by presenting results on surface parameterization, smoothing, remeshing, and thin-shell simulation.