Isometric Surfaces in Isotropic 3-Space

Abstract. Hard problems in Euclidean geometry can benefit from the solution of their counterparts in isotropic geometry, which is a structure preserving simplification of Euclidean geometry. For example, isotropic solutions may be turned into Euclidean ones via numerical optimization. The present paper is motivated by such a difficult problem, namely the design of flexible quad meshes. Flexible quad meshes constitute a special discretization of certain continuous isometric deformations. While the notion of isometric deformations of surfaces is straightforward in Euclidean space, a corresponding nontrivial notion in isotropic space has been missing and is the focus of this paper. By making Gauss’ Theorema Egregium a necessary condition, we develop a sensible notion of isometric surfaces in isotropic space. Well-known examples in Euclidean space, like isometries within the associated family of minimal surfaces, Bour’s theorem, and Minding isometries, find their natural analogues in isotropic space. We also include an extensive treatment of infinitesimal flexibility, or infinitesimal deformation, of surfaces. We prove results for the isotropic displacement diagrams in analogy to their well-known counterparts in Euclidean space, culminating in the existence of an isotropic Darboux wreath consisting of six surfaces. We show several interesting relations for special parametrizations involving Kœnigs and Voss nets of smooth and discrete surfaces within the Darboux wreath, and we encounter surfaces of constant Gaussian and mean curvature. Many of our results have an interpretation within statics. For example, the Airy stress surface associated to the static equilibrium of an inextensible membrane corresponds to the isotropic velocity diagram of an infinitesimally flexible surface.