In this talk we show how to compute weighted distance functions and geodesics on implicit surfaces, without the need to triangulate as commonly done in the literature. Based on the theory of geodesics
on Riemannian manifolds with boundaries, we prove that this can be done in a computationally optimal fashion and without affecting the basic accuracy of fast algorithms for computing geodesics on Euclidean spaces. The underlying algorithm becomes the same as the one for
Cartesian grids. This is joint work with Facundo Memoli.