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 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 briefly 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 case. We will emphasize the applicability of our discrete variational
approach to geometry for medical applications by showing results on surface
parameterization, smoothing, and remeshing, as well as thin-shell simulation.
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