Computing Riemannian Normal Coordinates on Triangle Meshes

• Imagine an ant walking around on the curved surface of a plant, a radio amateur planning to broadcast to a distant location across the globe or a pilot taking o from an airport - all of them are helped by egocentric maps of the world around them that shows directions and distances to various remote places. It is not surprising that this idea has already been used in cartography, where it is known as Azimuthal Equidistant Projection (AEP). If Earth is approximated by a sphere, distances and directions between two places are computed from arcs along great circles. In physics and mathematics, the same idea is known as Riemannian Normal Coordinates (RNC). It has been given a precise and general denition for surfaces (2-D), curved spaces (3-D) and generalized to smooth manifolds (N-D). RNC are the Cartesian coordinates of vectors that index points on the surface (or manifold) through the so called exponential map, which is a well known concept in dierential geometry. They are easily computed for a particular point if the inverse of the exponential map, the logarithm map, is known. Recently, RNC and similar coordinate systems have been used in computer graphics, visualization and related areas of research. In Fig. 1 for instance, RNC are used to produce a texture on the Stanford bunny through decal compositing. Given the growing use of RNC, which is further elaborated on in the next section, it is meaningful to develop accurate and reproducible techniques to compute this parameterization. In this paper, we describe a technique to compute RNC for surfaces represented by triangular meshes, which is the predominant representation of surfaces in computer graphics. The method that we propose has similarities to the Logmap framework, which has previously been developed for dimension reduction of unorganized point clouds in high-dimensional spaces, a.k.a. manifold learning. For this reason we sometimes refer to it as "Logmap for triangular meshes" or simply Logmap.

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