Schwerdtfeger-Fillmore-Springer-Cnops Construction Implemented in GiNaC

TL;DR

This paper details the implementation of the Schwerdtfeger-Fillmore-Springer-Cnops construction within GiNaC, enabling linearization of Moebius group actions in arbitrary dimensions with applications in geometry and engineering.

Contribution

It introduces a flexible, dimension-agnostic implementation of SFSCc using Clifford algebra in GiNaC, including 2D cycle visualization and potential porting to other CAS.

Findings

Implementation supports arbitrary dimensions and metrics

Includes visualization tools for 2D cycles

Serves as backbone for published mathematical results

Abstract

This paper presents an implementation of the Schwerdtfeger-Fillmore-Springer-Cnops construction (SFSCc) along with illustrations of its usage. SFSCc linearises the linear-fraction action of the Moebius group in R^n. This has clear advantages in several theoretical and applied fields including engineering. Our implementation is based on the Clifford algebra capacities of the GiNaC computer algebra system (http://www.ginac.de/), which were described in cs.MS/0410044. The core of this realisation of SFSCc is done for an arbitrary dimension of R^n with a metric given by an arbitrary bilinear form. We also present a subclass for two dimensional cycles (i.e. circles, parabolas and hyperbolas), which add some 2D specific routines including a visualisation to PostScript files through the MetaPost (http://www.tug.org/metapost.html) or Asymptote (http://asymptote.sourceforge.net/) packages. This software is the backbone of many results published in math.CV/0512416 and we use its applications their for demonstration. The library can be ported (with various level of required changes) to other CAS with Clifford algebras capabilities similar to GiNaC. There is an ISO image of a Live Debian DVD attached to this paper as an auxiliary file, a copy is stored on Google Drive as well.