Abstract
A mesh with planar faces is called an edge offset (EO) mesh if there exists a combinatorially equivalent mesh
such that corresponding edges of
and
lie on parallel lines of constant distance d. The edges emanating from a vertex of
lie on a right circular cone. Viewing
as set of these vertex cones, we show that the image of
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 4-space 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 Christoffel duality.
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Communicated by Rida Farouki.
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Pottmann, H., Grohs, P. & Blaschitz, B. Edge offset meshes in Laguerre geometry. Adv Comput Math 33, 45–73 (2010). https://doi.org/10.1007/s10444-009-9119-6
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DOI: https://doi.org/10.1007/s10444-009-9119-6
Keywords
- Discrete differential geometry
- Laguerre geometry
- Edge offset mesh
- Koebe polyhedron
- Minimal surface
- Laguerre minimal surface