Half-Turns and Clifford Configurations in the Inversive Plane

1977 ◽  
Vol s2-15 (3) ◽  
pp. 521-533 ◽  
Author(s):  
J. F. Rigby
Keyword(s):  
Inversive Plane

scholarly journals Symmetry in Regular Polyhedra Seen as 2D Möbius Transformations: Geodesic and Panel Domes Arising from 2D Diagrams

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 356 ◽  
Author(s):  
Jose Diaz-Severiano ◽  
Valentin Gomez-Jauregui ◽  
Cristina Manchado ◽  
Cesar Otero
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Space Structures ◽  
Inversive Plane ◽  
Delaunay Triangulations ◽  
Möbius Transformations ◽  
Regular Polyhedra ◽  
Complex Design ◽  
Elliptic Geometry ◽  
Mobius Transformations

This paper shows a methodology for reducing the complex design process of space structures to an adequate selection of points lying on a plane. This procedure can be directly implemented in a bi-dimensional plane when we substitute (i) Euclidean geometry by bi-dimensional projection of the elliptic geometry and (ii) rotations/symmetries on the sphere by Möbius transformations on the plane. These graphs can be obtained by sites, specific points obtained by homological transformations in the inversive plane, following the analogous procedure defined previously in the three-dimensional space. From the sites, it is possible to obtain different partitions of the plane, namely, power diagrams, Voronoi diagrams, or Delaunay triangulations. The first would generate geo-tangent structures on the sphere; the second, panel structures; and the third, lattice structures.

A Conformal Proof of a Jordan Curve Problem

1969 ◽  
Vol 12 (5) ◽  
pp. 673-674 ◽  
Author(s):  
G. Spoar ◽  
N.D. Lane
Keyword(s):  
Jordan Curve ◽  
Connected Region ◽  
Inversive Plane ◽  
Simply Connected Region ◽  
Simply Connected

The following theorem appears in [1].Let R be a closed simply connected region of the inversive plane bounded by a Jordan curve J, and let J be divided into three closed arcs A1, A2, A3. Then there exists a circle contained in R and having points in common with all three arcs.

Five Mutually Tangent Spheres and Various Associated Configurations

1982 ◽  
Vol 25 (2) ◽  
pp. 149-163
Author(s):  
J. F. Rigby
Keyword(s):  
Inversive Plane ◽  
Geometrical Progression

AbstractIf five spheres σ0, σ1, …, σ4 touch each other externally and have radii in geometrical progression, there is a dilative rotation mapping σ0, σ1, σ2, σ3, to σ1, σ2, σ3, σ4; the dilatation factor is shown to be negative. The ten points of contact of the spheres lie by fours on 15 circles, forming a (154106) configuration in inversive space. In the corresponding configuration in the inversive plane, the 15 circles meet again in 60 points, which lie by fours on 45 circles touching by threes at each of the 60 points, and forming a configuration isomorphic to that of 60 Pascal lines (associated with six points on a conic) meeting by fours at 45 points. The 45 circles arise from ten Money-Coutts configurations of nine anti-tangent cycles. Conjectures are made about other circles through the 60 points.

Uniqueness of the inversive plane of order 7

1973 ◽  
Vol 8 (1) ◽  
pp. 21-26 ◽  
Author(s):  
Ralph Hugh Francis Denniston
Keyword(s):  
Inversive Plane

A Beckman-Quarles Type Theorem for Coxeter's Inversive Distance

1991 ◽  
Vol 34 (4) ◽  
pp. 492-498 ◽  
Author(s):  
J. A. Lester
Keyword(s):  
Type Theorem ◽  
Möbius Transformation ◽  
Inversive Plane ◽  
The Real ◽  
Mobius Transformation

AbstractWe prove that a bijective transformation on the set of circles in the real inversive plane which preserves pairs of circles a fixed inversive distance ρ > 0 apart must be induced by a Möbius transformation.

The Inversive Plane

1941 ◽  
Vol 48 (9) ◽  
pp. 589 ◽  
Author(s):  
B. C. Patterson
Keyword(s):  
Inversive Plane
Sign in / Sign up

Export Citation Format

Share Document