Discrete S-Conical Scherk Tower
Alexander I. Bobenko, Tim Hoffmann, Benno König, Stefan Sechelmann
Media
Description
The classical Scherk surfaces were discovered by H.F. Scherk in [1]. For a comprehensive treatment of the Scherk minimal Surfaces see: [2]. We present a discrete version of the singly periodic Scherk surface, also known as Scherk's second minimal surface. It is a discrete s-conical version, see [3], of this surface, see [A Fundamental Piece] and the corresponding Gauss image [Discrete Gauss Map]. It is constructed using orthogonal circle patterns on the sphere (see [4]) to create a discrete version of the Gauss image.
In additional to this, we present data for the the discrete associate family. Scherk's discrete singly periodic minimal surface contains Scherk's doubly periodic surface at \(\gamma=\frac{\pi}{2}\) in the associate family, see [Conjugate Scherk Minimal Surface]. This surface is parameterized along asymptotic lines as in the smooth case.
Other digital versions of this model can be found at [5], and [6], and [7].
References
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Heinrich Ferdinand Scherk.
Bemerkungen über die kleinste Fläche innerhalb gegebener Grenzen.
Journal für Reine und Angewandte Mathematik, 1835(13):185–208, 1835.
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H. Karcher.
Construction of Minimal Surfaces.
Univ., 1989.
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Alexander Bobenko, Tim Hoffmann, Benno König, and Stefan Sechelmann.
Towards a unifying theory of discrete minimal surfaces.
in preparation, 2015.
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Alexander I. Bobenko and Boris A. Springborn.
Variational principles for circle patterns and Köbe’s theorem.
Trans. Amer. Math. Soc, 2004(356):659–689, 2004.
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3DXM Consortium.
Saddle Tower.
2004.
URL: http://virtualmathmuseum.org/Surface/saddle_tower/saddle_tower.html. -
Eric W. Weisstein.
Scherk's Minimal Surfaces.
2000.
URL: http://mathworld.wolfram.com/ScherksMinimalSurfaces.html. -
et. al. James T. Hoffman.
Scherk's Second Surface.
2002.
URL: http://www.msri.org/publications/sgp/jim/geom/minimal/library/scherk2/index.html.
