Isothermic triangulated surfaces

Abstract

Abstract. We found a class of triangulated surfaces in Euclidean space which have similar properties as isothermic surfaces in Differential Geometry. We call a surface isothermic if it admits an infinitesimal isometric deformation preserving the mean curvature integrand locally. We show that this class is Möbius invariant. Isothermic triangulated surfaces can be characterized either in terms of circle patterns or based on conformal equivalence of triangle meshes. This definition generalizes isothermic quadrilateral meshes.A consequence is a discrete analog of minimal surfaces. Here the Weierstrass data needed to construct a discrete minimal surface consist of a triangulated plane domain and a discrete harmonic function.

Vertex Normals and Face Curvatures of Triangle Meshes

“…(6) ("version 1") behaves differently from normality according to Eq. (6 * ) ("version 2"), while there is hardly any difference between conditions (6 * ) and (11). with χ = f + v − e as the Euler characteristic.…”

“…The theory presented in [2] is restricted to offset-like pairs of polyhedral surfaces where corresponding edges and faces are parallel. There are ongoing efforts to extend this theory to more general situations (we point to recent work on quad meshes [10] and on isothermic triangle meshes of constant mean curvature [11]). It is therefore remarkable that at least for the situation described here, triangle meshes allow an approach to curvatures and even a shape operator which is likewise guided by the Steiner formula, but without the rather restrictive property of parallelity (which for triangle meshes would be even more restrictive).…”

“…It turns out that there is no substantial difference between Eqs. (6 * ) and (11). Faces are colored according to the angle β enclosed between the congruence line at the barycenter and the face's normal there.…”

Holomorphic Vector Fields and Quadratic Differentials on Planar Triangular Meshes

“…The results on planar triangular meshes in this paper are closely related to isothermic triangulated surfaces in Euclidean space [8].…”

“…A similar formula for quadrilateral meshes with factorized real cross ratios was established by Bobenko and Pinkall [1]. Here we will use the definition of a discrete minimal surface f with Gauß map n given in [8]:…”