The Ribaucour families of discrete R-congruences

Rörig, Thilo; Szewieczek, Gudrun

doi:10.1007/s10711-021-00614-1

Cited by 6 publications

(17 citation statements)

References 28 publications

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“…For later reference, we also recall [2,30] that the formula for the cross-ratio of four spheres which are pairwise related by a Lie inversion simplifies: let s 1 , s 2 ∈ P(L), s 1 , s 2 = 0, be two spheres that are not contained in the linear sphere complex P(L ∩ {a} ⊥ ), then cr(s 2 , σ a (s 2 ), σ a (s 1 ),…”

Section: Definitionmentioning

confidence: 99%

“…Due to condition (iii), any two adjacent coordinate surfaces of the same family are related by a discrete Ribaucour transformation (cf [4,7,30]).…”

Section: Notationmentioning

confidence: 99%

“…The proposed construction will rely on the following general observations on discrete Ribaucour sphere congruences with a totally umbilic (discrete) envelope: Lemma 28 (cf [30]). Let r be a discrete Ribaucour sphere congruence that admits a totally umbilic envelope, then r is a (2, 1)-congruence and, on any face, the contact elements of the totally umbilic envelope coincide with contact elements of the corresponding R-Dupin cyclide along one circular curvature line of it.…”

Section: Corollary 27mentioning

confidence: 99%

“…The discrete Ribaucour sphere congruence r enveloped by (f, u) is then provided by the spheres in the contact elements of f that lie in the linear sphere complex P(L∩n ⊥ ). Hence, u can be expressed in terms of its (constant) curvature sphere n and the enveloped Ribaucour sphere congruence r by u i = span{n, r i } (cf [10,30]). To avoid useless case analyses, a totally umbilic discrete Legendre map with two families of circular curvature lines will also be called a discrete Dupin cyclide.…”

Section: Discrete Orthogonal Systems With Dupin Cyclides As Coordinat...mentioning

confidence: 99%

“…To begin with, we investigate the Ribaucour pair (f, u) and its Ribaucour sphere congruence along a fixed coordinate line of V. Since the spheres of the contact elements of f along each coordinate line all lie in a 3-dimensional projective subspace of P(R 4,2 ), the spheres of the enveloped Ribaucour congruence along each coordinate line are curvature spheres of another (constant) Dupin cyclide (see also [30]). Since f is a discrete Dupin cyclide, along this coordinate line, all contact elements F := {f m , f n , • • • , f z } share a common curvature sphere; we denote this sphere by s. Furthermore, all contact elements U := {u m , u n , • • • , u z } intersect in the constant sphere n (for a schematic see Figure 5 top).…”

Section: Theorem 32mentioning

confidence: 99%

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Discrete cyclic systems and circle congruences

Hertrich-Jeromin,

Szewieczek

2021

Preprint

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We discuss integrable discretizations of 3-dimensional cyclic systems, that is, orthogonal coordinate systems with one family of circular coordinate lines. In particular, the underlying circle congruences are investigated in detail, and characterized by the existence of a certain flat connection. Within the developed framework, discrete cyclic systems with a family of discrete flat fronts in hyperbolic space and discrete cyclic systems, where all coordinate surfaces are discrete Dupin cyclides, are investigated. MSC 2020. 53A70 (Primary) • 53A31 • 53A35

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Section: Definitionmentioning

confidence: 99%

“…Due to condition (iii), any two adjacent coordinate surfaces of the same family are related by a discrete Ribaucour transformation (cf [4,7,30]).…”

Section: Notationmentioning

confidence: 99%

Section: Corollary 27mentioning

confidence: 99%

Section: Discrete Orthogonal Systems With Dupin Cyclides As Coordinat...mentioning

confidence: 99%

Section: Theorem 32mentioning

confidence: 99%

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Discrete cyclic systems and circle congruences

Hertrich-Jeromin,

Szewieczek

2021

Preprint

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Discrete cyclic systems and circle congruences

Hertrich-Jeromin

Szewieczek

2022

Annali di Matematica

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We discuss integrable discretizations of 3-dimensional cyclic systems, that is, orthogonal coordinate systems with one family of circular coordinate lines. In particular, the underlying circle congruences are investigated in detail and characterized by the existence of a certain flat connection. Within the developed framework, discrete cyclic systems with a family of discrete flat fronts in hyperbolic space and discrete cyclic systems, where all coordinate surfaces are discrete Dupin cyclides, are investigated.

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Meshes with Spherical Faces

Kilian,

Cisneros,

Müller

et al. 2023

ACM Trans. Graph.

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Discrete surfaces with spherical faces are interesting from a simplified manufacturing viewpoint when compared to other double curved face shapes. Furthermore, by the nature of their definition they are also appealing from the theoretical side leading to a Möbius invariant discrete surface theory. We therefore systematically describe so called sphere meshes with spherical faces and circular arcs as edges where the Möbius transformation group acts on all of its elements. Driven by aspects important for manufacturing, we provide the means to cluster spherical panels by their radii. We investigate the generation of sphere meshes which allow for a geometric support structure and characterize all such meshes with triangular combinatorics in terms of non-Euclidean geometries. We generate sphere meshes with hexagonal combinatorics by intersecting tangential spheres of a reference surface and let them evolve - guided by the surface curvature - to visually convex hexagons, even in negatively curved areas. Furthermore, we extend meshes with circular faces of all combinatorics to sphere meshes by filling its circles with suitable spherical caps and provide a remeshing scheme to obtain quadrilateral sphere meshes with support structure from given sphere congruences. By broadening polyhedral meshes to sphere meshes we exploit the additional degrees of freedom to minimize intersection angles of neighboring spheres enabling the use of spherical panels that provide a softer perception of the overall surface.

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The Ribaucour families of discrete R-congruences

Cited by 6 publications

References 28 publications

Discrete cyclic systems and circle congruences

Discrete cyclic systems and circle congruences

Discrete cyclic systems and circle congruences

Meshes with Spherical Faces

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