3.2, we then take a different point of view: we start from a discrete 2-dimensional family of circles and investigate under what conditions there exists a family of orthogonal discrete Legendre maps that gives rise to a discrete cyclic system. Here we approach the discretization of smooth cyclic systems from two different angles: firstly, we consider discrete triply orthogonal systems that contain two coordinate surface families of discrete channel surfaces and, therefore, have a family of circular orthogonal trajectories. 2 on basic concepts and facts on circles and spheres that will be essential for what follows, the main notions of the text are presented in Sect. In this way, we hope to contribute to a methodologically systematic and transparent approach to the field. Here, we aim to present a more explicit treatment and, in particular, to explicitly investigate and employ the properties of connections built from Lie and M-Lie inversions in the context of cyclic systems. In a sometimes more implicit way, such connections have been used in the theory of discrete orthogonal systems for a long time, see. ![]() In our case, these will be Lie and M-Lie inversions (see Def 1) that can be used to generate cyclic circle congruences and their orthogonal nets in an efficient way. Moreover, since the established theory naturally generalizes to higher-dimensional systems, it shall lead to discrete notions for 3-dimensional conformally flat hypersurfaces and Möbius flat hypersurfaces as orthogonal surfaces of discrete cyclic systems stemming from discrete flat fronts.Īnother main goal of this paper is to further examine, and promote, the use of discrete connections that are given in the simplest way possible: by the “reflections” of the underlying ambient geometry. 4.2, we shall investigate relations of various approaches to flat fronts in hyperbolic space and, in particular, prove the existence of a Weierstrass-type representation for discrete flat fronts in. For example, based on the observations in Sect. In this way, we anticipate to pave the way for further studies in this context, inspired by the rich smooth theory as sketched above. The main goal of the present work is to explore discrete counterparts of cyclic circle congruences and their associated cyclic coordinate systems (see Def 11 and 16), based on this definition. Īn integrable discretization of orthogonal coordinate systems was given in where those were introduced as higher-dimensional circular (principal) nets. Higher dimensional analogues lead to 3-dimensional conformally flat hypersurfaces and, more generally, Möbius flat hypersurfaces. The former example generalizes to a remarkable class of cyclic circle congruences given by curved flats in the space of circles (so-called flat spherical or hypercyclic systems) that come with an orthogonal family of Guichard surfaces. Among them are, for example, pseudospherical surfaces related by Bianchi transformations or parallel families of flat fronts in hyperbolic space. ![]() Secondly, cyclic circle congruences can be employed to construct (families of) surfaces of various special types by imposing further (geometric) conditions on the cyclic circle congruence. Cyclic systems are widely used in physics and include rotational systems, as for example, spherical and toroidal coordinates. Examples are provided by orthogonal systems with a family of surfaces that are parallel in a fixed space form (cf ) and special cyclidic coordinate systems (also called totally cyclic), where all coordinate surfaces are Dupin cyclides. As a consequence, two families of coordinate surfaces then consist of channel surfaces. The main motivations to study these special circle congruences seem to be twofold: firstly, a surface family orthogonal to a cyclic circle congruence gives rise to a special orthogonal coordinate system ( cyclic system), where the orthogonal trajectories of one family are circular. Many other cyclidic geometries can be obtained by studying R-separation of variables for the Laplace equation.In, Ribaucour investigated circle congruences that admit a 1-parameter family of orthogonal surfaces and called those congruences cyclic (also called normal). In Maxime Bôcher's 1891 dissertation, Ueber die Reihenentwickelungen der Potentialtheorie, it was shown that the Laplace equation in three variables can be solved using separation of variables in 17 conformally distinct quadric and cyclidic coordinate geometries. įamilies of cyclides give rise to various cyclidic coordinate geometries. Where Q is a 3x3 matrix, P and R are a 3-dimensional vectors, and A and B are constants. ![]() There are several equivalent definitions of Dupin cyclides.
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