Some time ago I wrote a blog post about geodesics and different methods how they could be computed in Houdini. Geodesic distances are great since they represent the distance, or more precisely, the shortest distance between points on a surface. While this is an important property and useful for many things we might run into a situation in which other properties are even more important, for instance smoothness. This might be the case if we want to generate a smooth vector field based on the distance or if we are working on meshes with holes for example. However, a nice solution to these problems is to compute biharmonic distances instead of geodesic distances. Biharmonic distance is not necessarily the true distance between points on a surface but it is smooth and globally shape-aware.
Contrary to geodesic distance, the biharmonic distance is not computed directly on the surface but on the eigenvectors of the laplacian. Basically it’s the average distance between pairwise points projected onto incrementally increasing eigenvectors. The general accuracy is therefore highly dependent on the number of computed eigenvectors – the more the better. If we need to get exact distances we have to compute all eigenvectors or in other words, we need to perform the computationally rather expensive full eigendecomposition of the laplacian. For a sufficient approximation however, we usually can get away with the first 20-100 eigenvectors depending on mesh density and shape. For the implementation in Houdini I’m using Spectra and Eigen together with the HDK. Even though it’s quite fast to perform the eigenvalue/eigenvector decomposition, the computation of geodesic distances is of course much faster, especially on large meshes.
Y. Lipman, R. M. Rustamov, T. A. Funkhouser: Biharmonic Distance