We propose a discrete surface theory in ℝ3 that unites the most prevalent versions of discrete special parametrizations. Our theory encapsulates a large class of discrete surfaces given by a Lax representation and, in particular, the one-parameter associated families of constant curvature surfaces. Our theory is not restricted to integrable geometries, but extends to a general surface theory. A quad net is a map from a strongly regular polytopal cell decomposition of a surface with all faces being quadrilaterals into ℝ3 with nonvanishing straight edges. A polytopal cell decomposition is strongly regular if each edge connects distinct vertices and meets at most two faces. Notice, in particular, that nonplanar faces are admissible. In discrete differential geometry, quad nets are understood as discretizations of parametrized surfaces [9, 10, 13]. In this agenda many classes of special surfaces have been discretized using algebro-geometric approaches for integrable geometry—originally using discrete analogues of soliton theory techniques (e.g., discrete Lax pairs and finite-gap integration [6]) to construct nets, but more recently using the notion of 3D consistency (reviewed in [10]). As in the smooth setting, these approaches have been successfully applied to space forms (see, e.g., [15–17, 22]). As an example consider the case of K-surfaces, i.e., surfaces of constant negative Gauß curvature. The integrability equations of classical surface theory are equivalent to the famous sine-Gordon equation [1, 20]. In an integrable discretization, the sine-Gordon equation becomes a finite difference equation for which integrability is encoded by a certain closing condition around a 3D cube. Both in the smooth and discrete settings, integrability is bound to specific choices of parameterizations, such as asymptotic line parametrizations for K-surfaces. In this way different classes of surfaces, such as minimal surfaces or surfaces of constant mean curvature (CMC), lead to different partial differential equations and give rise to different parametrizations. In the discrete case, this is reflected by developments that treat different special surfaces by disparate approaches. These integrable discretizations maintain characteristic properties of their smooth counterparts (e.g., the transformation theory of Darboux, Bäcklund, Bianchi, etc.). What has been lacking, however, is a unified discrete theory that lifts the restriction of special surface parametrizations. Indeed, different from the case of classical smooth surface theory, existing literature does not provide a general discrete theory for quad nets.
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Sep 1, 2017
Guido Kreis 
Open Access