Hyperbolic barycentric coordinates and applications

Abstract

Barycentric coordinates are a fundamental tool in computer graphics and geometry processing. A variety of ways has been proposed for constructing such coordinates on the Euclidean plane. The spherical barycentric coordinates are also developed. This paper completes this construction for the hyperbolic plane case. We define hyperbolic barycentric coordinates (HBC) that describe the position of a point in the hyperbolic plane with respect to the vertices of a given geodesic polygon. We construct explicitly three kinds of HBC, namely hyperbolic Wachspress, mean values and discrete harmonic coordinates. These coordinates have properties which resemble those of the planar ones, and they are invariant by the Lorentzian transformations. Furthermore, we figure out the HBC on the Poincaré disk model. The HBC associated to a point in a hyperbolic triangle are unique. We develop two expressions of these coordinates, taking into account the parameters of a point inside the triangle.In addition, we exploit these coordinates to define a parameterization of a surface-mesh with boundary into the Poincaré disk, and we show some examples. This hyperbolic parameterization extends that of the planar one, known as Tutte's embedding.Furthermore, we demonstrate the efficiency of these coordinates by giving other applications. Namely hyperbolic deformation and rapid shape morphing.