Schwerdtfeger-Fillmore-Springer-Cnops Construction Implemented in GiNaC

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.Comment: LaTeX, 82 p; 11 PS graphics in two figures, the full source files and ISO image of Live DVD are included; v9: library update for the book on Moebius transformations; v10: an ISO image of a Live DVD is attached to the paper; v11: a bug is fixed; v12: Library is uupdated, the reference to a larger project is adde

Conformal Parametrisation of Loxodromes by Triples of Circles

We provide a parametrisation of a loxodrome by three specially arranged cycles. The parametrisation is covariant under fractional linear transformations of the complex plane and naturally encodes conformal properties of loxodromes. Selected geometrical examples illustrate the usage of parametrisation. Our work extends the set of objects in Lie sphere geometry---circle, lines and points---to the natural maximal conformally-invariant family, which also includes loxodromes.Comment: 14 pages. 9 PDF in four figures, AMS-LaTe

Two-Dimensional Conformal Models of Space-Time and Their Compactification

We study geometry of two-dimensional models of conformal space-time based on the group of Moebius transformation. The natural geometric invariants, called cycles, are used to linearise Moebius action. Conformal completion of the space-time is achieved through an addition of a zero-radius cycle at infinity. We pay an attention to the natural condition of non-reversibility of time arrow in order to get a correct compactification in the hyperbolic case.Comment: 8 pages,AMS-LaTeX, 18 PS figures; v2--small corrections; v3--add two coments on notations and multidimensional generalisation

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

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 Möbius–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 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

Cycles Cross Ratio: an Invitation

The paper introduces cycles cross ratio, which extends the classic cross ratio of four points to various settings: conformal geometry, Lie spheres geometry, etc. Just like its classic counterpart cycles cross ratio is a measure of anharmonicity between spheres with respect to inversion. It also provides a M\"obius invariant distance between spheres. Many further properties of cycles cross ratio awaiting their exploration. In abstract framework the new invariant can be considered in any projective space with a bilinear pairing.Comment: 10 pages, 3 PDF files in two figures, AMS-LaTeX; v2: Steiner power is linked to cross-ratio; v3: Jupyter notebook linked; v4: numerous improvements; v5: a couple of references is added; v6: minor correction

Erlangen Program at Large-2.5: Induced Representations and Hypercomplex Numbers

In the search for hypercomplex analytic functions on the half-plane, we review the construction of induced representations of the group G=SL(2,R). Firstly we note that G-action on the homogeneous space G/H, where H is any one-dimensional subgroup of SL(2,R), is a linear-fractional transformation on hypercomplex numbers. Thus we investigate various hypercomplex characters of subgroups H. The correspondence between the structure of the group SL(2,R) and hypercomplex numbers can be illustrated in many other situations as well. We give examples of induced representations of SL(2,R) on spaces of hypercomplex valued functions, which are unitary in some sense. Raising/lowering operators for various subgroup prompt hypercomplex coefficients as well. The paper contains both English and Russian versions. Keywords: induced representation, unitary representations, SL(2,R), semisimple Lie group, complex numbers, dual numbers, double numbers, Moebius transformations, split-complex numbers, parabolic numbers, hyperbolic numbers, raising/lowering operators, creation/annihilation operatorsComment: LaTeX2e; 17 pp + 13 pp of a source code; 5 EPS pictures in two Figures; v2: minor improvements and corrections; v3: a section on raising/lowering operators is added; v4: typos are fixed; v5: Introduction is added, open problems are expanded.v6: Russian translation is added, references areupdated, NoWeb and C++ source codes are added as ancillary files. arXiv admin note: substantial text overlap with arXiv:0707.402

Poincaré extension of Möbius transformations

Given sphere preserving (Möbius) transformations in n-dimensional Euclidean space one can use the Poincaré extension to obtain sphere preserving transformations in a half-space of n+1 dimensions. The Poincaré extension is usually provided either by an explicit formula or by some geometric construction. We investigate its algebraic background and describe all available options. The solution is given in terms of one-parameter subgroups of Möbius transformations acting on triples of quadratic forms. To focus on the concepts, this paper deals with the Möbius transformations of the real line only