An extension of Mobius--Lie geometry with conformal ensembles of cycles and its implementation in a GiNaC library

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 M\"obius--Lie geometry. It was recently demonstrated in several related papers, that such ensembles of cycles naturally parameterize many other conformally-invariant families of objects, e.g. loxodromes or continued fractions.
 The paper describes a method, which reduces a collection of conformally in\-vari\-ant 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 {\CPP} 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.

MoebInv: C++ libraries for manipulations in non-Euclidean geometry

“…The package accumulates more than 15 years of work and is still under active development. The design of the library figure shaped the general theoretical approach to the extension of Möbius-Lie geometry [2,3], which leaded to specific realisations in [33,34,35]. Furthermore, it shall be helpful for computer experiments in Lie sphere geometry of indefinite or nilpotent metrics since our intuition is not elaborated there in contrast to the Euclidean space [22,26,36].…”

“…New results, e.g. elliptic and hyperbolic conformal version of the nine-point theorem, discovered with the second library figure were published in [2,3], cf. Fig.…”

“…As a non-trivial example of automated proof accomplished by the figure library for the first time, we present a FLT-invariant version of the classical nine-point theorem [2,3], which is illustrated in the attached video.…”

“…A FLT-invariant relations between cycles include incidence, orthogonality, tangency, etc. The efficiency of the package MoebInv is based on the mathematical formalism [2,13] which encodes the variety of different FLT-invariant relations between cycles in a unified way. Besides aesthetical value the conformal and Möbius-Lie geometries have countless applications ranging from quantum spin dynamics [14], cosmology [15] and integrable systems [16] to computer-aided design [17] and physiological models [18].…”

Möbius--Lie Geometry and Its Extension

“…An alternative demonstration uses three observations: Finally, expression (15), as well as (16)-(17) below, can be calculated by the specialised CAS for Möbius invariant geometries [41,49]. This standard approach can be widened as follows.…”