Geodesic Polar Coordinates (GPCs) on a smooth surface S are local surface coordinates that relates a surface point to a planar parameter point by the length and direction of a corresponding geodesic curve on S. They are intrinsic to the surface and represent a natural local parameterization with useful properties. We present a simple and efficient algorithm to approximate GPCs on both triangle ...

We present a brief but nearly self-contained proof of a formula for the Weil-Petersson Hessian of the geodesic length of a closed curve (either simple or not simple) on a hyperbolic surface. The formula is the sum of the integrals of two naturally defined positive functions over the geodesic, proving convexity of this functional over Teichmuller space (due to Wolpert (1987)). We then estimate t...

We study properties of Sobolev-type metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolev-type metrics with constant coefficients of order 2 and higher is globally well-posed for smooth initial data as well as for initial data in certain Sobolev spaces. Thus the space of closed plane curves equipped with such a metric is geodesically complete. We find low...

In a non-complete graph $Gamma$, a vertex triple $(u,v,w)$ with $v$ adjacent to both $u$ and $w$ is called a $2$-geodesic if $uneq w$ and $u,w$ are not adjacent. The graph $Gamma$ is said to be   $2$-geodesic transitive if its automorphism group is transitive on arcs, and also on 2-geodesics. We first produce a reduction theorem for the family of $2$-geodesic transitive graphs of prime power or...

This paper describes a shape signature that captures the intrinsic geometric structure of 3D objects. The primary motivation of the proposed approach is to encode a 3D shape into a one-dimensional geodesic distribution function. This compact and computationally simple representation is based on a global geodesic distance defined on the object surface, and takes the form of a kernel density esti...

We consider the Brezis–Nirenberg problem for the Laplacian with a singular drift for a (geodesic) ball in both R and S, 3 ≤ n ≤ 5. The singular drift we consider derives from a potential which is symmetric around the center of the (geodesic) ball. Here the potential is given by a parameter (δ say) times the logarithm of the distance to the center of the ball. In both cases we determine the exac...

The abstract boundary has, in recent years, proved a general and flexible way to define the singularities of space-time. In this approach an essential singularity is a nonregular boundary point of an embedding which is accessible by a chosen family of curves within finite parameter distance. Ashley and Scott proved the first theorem relating essential singularities in strongly causal space-time...

In this paper, we present an automatic 3D face recognition system. This system is based on the representation of human faces surfaces as collections of Iso-Geodesic Curves (IGC) using 3D Fast Marching algorithm. To compare two facial surfaces, we compute a geodesic distance between a pair of facial curves using a Riemannian geometry. In the classifying step, we use: Neural Networks (NN), K-Near...

Here, we intend to introduce an invariant bipolar representation for curved surfaces based on the superposition of the two geodesic potentials defined from two given reference points on the surface. By considering a levels set of such geodesic potentials, a finite set of invariant points under 3D motion group is obtained. We propose to compare two different numerical algorithms which compute th...

In this paper, we propose a novel Patch Geodesic Distance (PGD) to transform the texture map of an object through its shape data for robust 2.5D object recognition. Local geodesic paths within patches and global geodesic paths for patches are combined in a coarse to fine hierarchical computation of PGD for each surface point to tackle the missing data problem in 2.5D images. Shape adjusted text...