Improvements to Snap-Together Algebraic Surfaces

Abstract

Our continuing work is to make assemblable models of algebraic surfaces. We create our art aiming to represent the surfaces with as much accuracy as possible, which is difficult due to computing directions of plugs and sockets, thickening around singularities, and meshes being jagged and armadillo-like. Our first concern has been computing the medial axis for surfaces in 3D which will help with thickening and give us a new way to compute directions for plugs and sockets. Secondly, to attempt to get around the thickening issue we are using paper craft to create sample models. Thirdly, we have been working to produce a mesh which will be as smooth as possible to reflect the smoothness of the computed algebraic surfaces. The approach we have used for computing the medial axis involves using the Voronoi diagram and Delaunay triangulation. We used the application Pepakura to split and unfold models and we assembled them using paper. To improve mesh smoothness, we have been developing new triangulation algorithms to implement in Bertini_real that produce triangles which are as close to equilateral as possible. We have implemented a variety of triangulation algorithms to varying degrees of success.