Abstract:
A mesh M with planar faces is called an edge offset (EO) mesh if there exists a combinatorially equivalent mesh Md such that corresponding edges of M and Md lie on parallel lines of constant distance d. The edges emanating from a vertex of M lie on a right circular cone. Viewing M as set of these vertex cones, we show that the image of M under any Laguerre transformation is again an EO mesh. As a generalization of this result, it is proved that the cyclographic mapping transforms any EO mesh in a hyperplane of Minkowksi 4space into a pair of Euclidean EO meshes. This result leads to a derivation of EO meshes which are discrete versions of Laguerre minimal surfaces. Laguerre minimal EO meshes can also be constructed directly from certain pairs of Koebe meshes with help of a discrete Laguerre geometric counterpart of the classical Christoel duality.
Bibtex:
@article{pottmann2010eolag,
author = "Helmut Pottmann and Philipp Grohs and Bernhard Blaschitz",
title = "Edge offset meshes in {L}aguerre geometry",
journal = "Adv. Comp. Math.",
year = 2010,
volume = 33,
pages = "4573",
url = "http://www.dmg.tuwien.ac.at/grohs/papers/eolag.pdf",
doi = "http://dx.doi.org/10.1007/s1044400991196",
}

