A geodesic is a parameterized curve on a Riemannian manifold governed by a second order partial differential equation. Geodesics are notoriously unstable: small perturbations of the underlying manifold may lead to dramatic changes of the course of a geodesic. Such instability makes it difficult to use geodesics in many applications, in particular in the world of discrete geometry. In this paper, we consider a geodesic as the indicator function of the set of the points on the geodesic. From this perspective, we present a new concept called fuzzy geodesics and show that fuzzy geodesics are stable with respect to the Gromov-Hausdorff distance. Based on fuzzy geodesics, we propose a new object called the intersection configuration for a set of points on a shape and demonstrate its effectiveness in the application of finding consistent correspondences between sparse sets of points on shapes differing by extreme deformations.

@article{Sun:2010:FGA,
   author = "Jian Sun and Xiaobai Chen and Thomas Funkhouser",
   title = "Fuzzy Geodesics and Consistent Sparse Correspondences For Deformable
      Shapes",
   journal = "Computer Graphics Forum (Symposium on Geometry Processing)",
   year = "2010",
   month = jul,
   volume = "29",
   number = "5"
}