Ball-Morph: Definition, Implementation, and Comparative Evaluation

We define b-compatibility for planar curves and propose three ball morphing techniques between pairs of b-compatible curves. Ball-morphs use the automatic ball-map correspondence, proposed by Chazal et al. [1], from which we derive different vertex trajectories (linear, circular, and parabolic). All three morphs 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 ball-morphs and compare them to each other and other simple morphs (linear-interpolation, closest-projection, curvature-interpolation, Laplace-blending, and heat-propagation) using six cost measures (travel-distance, distortion, stretch, local acceleration, average squared mean curvature, and maximum squared mean curvature). The results depend heavily on the input curves. Nevertheless, we found that the linear ball-morph has consistently the shortest travel-distance and the circular ball-morph has the least amount of distortion.