An Extension of Moebius--Lie Geometry with Conformal Ensembles of Cycles and Its Implementation in a GiNaC Library

TL;DR

This paper introduces an extended Moebius--Lie geometric framework using ensembles of cycles interconnected by conformal relations, with a C++ library implementation supporting symbolic and numeric computations in various dimensions.

Contribution

It extends Moebius--Lie geometry to ensembles of cycles, providing a linear system reduction method and a versatile C++ library with Python interface for conformal geometric computations.

Findings

The method simplifies conformal relations to linear equations.

The library supports arbitrary dimensions and signatures.

Visualizations and animations are available for 2D and 3D cases.

Abstract

We propose to consider ensembles of cycles (quadrics), which are interconnected through conformal-invariant geometric relations (e.g. "to be orthogonal", "to be tangent", etc.), as new objects in an extended Moebius--Lie geometry. It was recently demonstrated in several related papers, that such ensembles of cycles naturally parameterise many other conformally-invariant objects, e.g. loxodromes or continued fractions. The paper describes a method, which reduces a collection of conformally invariant geometric relations to a system of linear equations, which may be accompanied by one fixed quadratic relation. To show its usefulness, the method is implemented as a C++ library. It operates with numeric and symbolic data of cycles in spaces of arbitrary dimensionality and metrics with any signatures. Numeric calculations can be done in exact or approximate arithmetic. In the two- and three-dimensional cases illustrations and animations can be produced. An interactive Python wrapper of the library is provided as well.