scholarly journals Hyperbolization of Euclidean Ornaments

10.37236/78 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Martin von Gagern ◽  
Jürgen Richter-Gebert
Keyword(s):  
Differential Geometry ◽  
Pattern Recognition ◽  
Hyperbolic Plane ◽  
Discrete Differential Geometry ◽  
Fundamental Region ◽  
Symmetry Groups ◽  
Symmetry Detection ◽  
Conformal Deformation ◽  
Optimal Resolution ◽  
Wallpaper Groups

In this article we outline a method that automatically transforms an Euclidean ornament into a hyperbolic one. The necessary steps are pattern recognition, symmetry detection, extraction of a Euclidean fundamental region, conformal deformation to a hyperbolic fundamental region and tessellation of the hyperbolic plane with this patch. Each of these steps has its own mathematical subtleties that are discussed in this article. In particular, it is discussed which hyperbolic symmetry groups are suitable generalizations of Euclidean wallpaper groups. Furthermore it is shown how one can take advantage of methods from discrete differential geometry in order to perform the conformal deformation of the fundamental region. Finally it is demonstrated how a reverse pixel lookup strategy can be used to obtain hyperbolic images with optimal resolution.

scholarly journals Modeling Membrane Dynamics in 3D using Discrete Differential Geometry

2021 ◽  
Vol 120 (3) ◽  
pp. 46a
Author(s):  
Cuncheng Zhu ◽  
Christopher T. Lee ◽  
Ravi Ramamoorthi ◽  
Padmini Rangamani
Keyword(s):  
Differential Geometry ◽  
Membrane Dynamics ◽  
Discrete Differential Geometry

Frieze Patterns in the Hyperbolic Plane

1974 ◽  
Vol 17 (1) ◽  
pp. 45-50 ◽  
Author(s):  
C. W. L. Garner
Keyword(s):  
Hyperbolic Plane ◽  
Euclidean Plane ◽  
Symmetry Groups ◽  
Euclidean Case ◽  
Parallel Lines

AbstractIt is well known that in the Euclidean plane there are seven distinct frieze patterns, i.e. seven ways to generate an infinite design bounded by two parallel lines. In the hyperbolic plane, this can be generalized to two types of frieze patterns, those bounded by concentric horocycles and those bounded by concentric equidistant curves. There are nine such frieze patterns; as in the Euclidean case, their symmetry groups are and

Automated Symmetry Exploitation in Engineering Analysis

2004 ◽  
Author(s):  
Krishnan Suresh
Keyword(s):  
Irreducible Representations ◽  
Point Symmetry ◽  
Symmetry Groups ◽  
Engineering Analysis ◽  
Symmetry Detection ◽  
Important Concept ◽  
Improve Accuracy ◽  
Speed Up ◽  
Boundary Mapping ◽  
Cell Construction

It is well known that one can exploit symmetry to speed-up engineering analysis and improve accuracy, at the same time. Not surprisingly, most CAE systems have standard ‘provisions’ for exploiting symmetry. However, these provisions are inadequate in that they needlessly burden the design engineer with time consuming and error-prone tasks of symmetry detection, symmetry cell construction and reformulation. In this paper, we propose and discuss an automated methodology for symmetry exploitation. First, we briefly review the theory of point symmetry groups that symmetry exploitation rests on. We then address symmetry detection and ‘symmetry cell’ construction. We then address an important concept of boundary mapping of symmetry cells, and relate it to the irreducible representations of point symmetry groups. By formalizing these concepts, we show how automated symmetry exploitation can be achieved, and discuss an implementation of the proposed work within the FEMLAB CAE environment.

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