Discrete Differential Geometry for $C^{1,1}$ Hyperbolic Surfaces of Non-Constant Curvature

Abstract: We develop a discrete differential geometry for surfaces of non-constant negative curvature, which can be used to model various phenomena from the growth of flower petals to marine invertebrate swimming. Specifically, we derive and numerically integrate a version of the classical Lelieuvre formulas that apply to immersions of $C^{1,1}$ hyperbolic surfaces of non-constant curvature. In contrast to the constant curvature case, these formulas do not provide an explicit method for constructing an immersion but rather describe an immersion via an implicit set of equations. We propose an iterative method for resolving these equations. Because we are interested in scenarios where the curvature is a function of the intrinsic material coordinates, in particular, on the geodesic distance from an origin, we suggest a fast marching method for computing geodesic distance on manifolds. We apply our methods to generate surfaces of non-constant curvature and demonstrate how one can introduce branch points to account for the sort of multi-generational buckling and subwrinkling observed in many applications.