Real-time visualization of Moebius transformations in space using Quaternionic-Bezier approach

Abstract

Moebius transformations in space are much more sophisticated than the classical case on the plane, which has been well studied. We present a WebGL approach for visualization of Moebius transformations in 3-space by animating deformations of geometric objects composed of patches parametrized by Quaternionic-Bezier formulas. The idea is to represent Moebius transformations in a quaternionic form as well, and to use GPU shaders for transforming control points, weights, and normals, then seamlessly stitching patches with different levels of detail, and computing points on every patch. Finally, we demonstrate the main classes of Moebius transformations in space on several 3D objects including primitive shapes, Dupin cyclide patchworks, and Utah Teapot.