Visualization of Darboux Cyclides

Abstract

Darboux cyclides are algebraic surfaces that can be interpreted as an intersection of the Möbius-quadric with another quadric in four-dimensional projective space. Based on the existence of coverings by circles of Darboux cyclides, various geometricalmaps between circles are defined, which induce polygon meshes. Using the Python librarypyddg, these polygon meshes are visualized in the open-source graphics software Blenderas surfaces. pyddg is an implementation of geometrical structures following the Erlangenprogram, which was initially released in 2022 by the geometry group at TU-Berlin. Selected parts of the visualization methods are being made available by embedding them into thelibrary. Pottmann et al. showed that certain types of Darboux cyclides carry 3-webs of circles. Webs are configurations of curves on topological surfaces, which are invariant under homeomorphisms. For 3-webs on smooth Darboux cyclides, we prove an inequality that could berelevant for proving the convergence of algorithms for closing such webs.