Infinitesimally flexible meshes and discrete minimal surfaces

Abstract:

We explore the geometry of isothermic meshes, conical meshes, and asymptotic meshes around the Christoffel dual construction of a discrete minimal surface. We present a discrete Legendre transform which realizes discrete minimal surfaces as conical meshes. Conical meshes turn out to be infinitesimally flexible if and only if their spherical image is isothermic, which implies that discrete minimal surfaces constructed in that way are infinitesimally flexible, and therefore possess reciprocal-parallel meshes. Those are discrete minimal surfaces in their own right. In our study of relative kinematics of infinitesimally flexible meshes we encounter characterizations of flexibility and isothermicity which are of incidence-geometric nature and are related to the classical Desargues configuration. The Lelieuvre formula for asymptotic meshes leads to another characterization of isothermic meshes in the sphere which is based on triangle areas.

Bibtex:

@article{wallner-2007-min, title = "Infinitesimally flexible meshes and discrete minimal surfaces", author = "Johannes Wallner and Helmut Pottmann", journal = {Monatshefte Math.}, year = 2007, note = "to appear", url = "http://www.geometrie.tugraz.at/wallner/cmin.pdf", }