| 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 this way are infinitesimally flexible, and therefore possess reciprocal-parallel
meshes. These 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. |
