b-morphs between b-compatible curves in the plane

Abstract

We define b-compatibility for planar curves and propose three ball morphing techniques (b-morphs) between pairs of b-compatible curves. B-morphs use the automatic ball-map correspondence, proposed by Chazal et al. [12], from which they derive vertex trajectories (Linear, Circular, Parabolic). All are symmetric, meeting both curves with the same angle, which is a right angle for the Circular and Parabolic. We provide simple constructions for these b-morphs using the maximal disks in the finite region bounded by the two curves. We compare the b-morphs to each other and to other simple morphs (Linear Interpolation (LI), Closest Projection (CP), Curvature Interpolation (CI), Laplace Blending (LB), Heat Propagation (HP)) using seven measures of quality deficiency (travel distance, distortion, stretch, local acceleration, surface area, average curvature, maximal curvature). We conclude that the ratios of these measures depends heavily on the test case, especially for LI, CI, and LB, which compute correspondence from a uniform geodesic parameterization. Nevertheless, we found that the Linear b-morph has consistently the shortest travel distance and that the Circular b-morph has the least amount of distortion.