An Excursion Through Discrete Differential Geometry

2020 ◽  
2021 ◽  
Vol 120 (3) ◽  
pp. 46a
Author(s):  
Cuncheng Zhu ◽  
Christopher T. Lee ◽  
Ravi Ramamoorthi ◽  
Padmini Rangamani

10.37236/78 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Martin von Gagern ◽  
Jürgen Richter-Gebert

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.


2021 ◽  
Vol 143 (8) ◽  
Author(s):  
Steven W. Grey ◽  
Fabrizio Scarpa ◽  
Mark Schenk

Abstract Origami-inspired approaches to deployable or morphing structures have received significant interest. For such applications, the shape of the origami structure must be actively controlled. We propose a distributed network of embedded actuators which open/close individual folds and present a methodology for selecting the positions of these actuators. The deformed shape of the origami structure is tracked throughout its actuation using local curvatures derived from discrete differential geometry. A Genetic Algorithm (GA) is used to select an actuation configuration, which minimizes the number of actuators or input energy required to achieve a target shape. The methodology is applied to both a deployed and twisted Miura-ori sheet. The results show that designing a rigidly foldable pattern to achieve shape-adaptivity does not always minimize the number of actuators or input energy required to reach the target geometry.


2006 ◽  
pp. 653-728 ◽  
Author(s):  
Alexander Bobenko ◽  
Richard Kenyon ◽  
John Sullivan ◽  
Günter Ziegler

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