Time‐Discrete Geodesics in the Space of Shells

Abstract

Building on concepts from continuum mechanics, we offer a computational model for geodesics in the space of thin shells, with a metric that reflects viscous dissipation required to physically deform a thin shell. Different from previous work, we incorporate bending contributions into our deformation energy on top of membrane distortion terms in order to obtain a physically sound notion of distance between shells, which does not require additional smoothing. Our bending energy formulation depends on the so-called relative Weingarten map, for which we provide a discrete analogue based on principles of discrete differential geometry. Our computational results emphasize the strong impact of physical parameters on the evolution of a shell shape along a geodesic path.

Elastic 3D shape analysis using square-root normal field representation

“…The first one is inspired by the elasticity theory in physics [30] where surfaces are treated as thin shells, i.e. a thin threedimensional material of thickness δ. Stretching in this case is caused by in-layer (tangential) shear or compression while bending is caused by friction due to transversal shear [15].…”

“…If geodesics, mean shapes, PCA, etc. could be computed in Q under the L 2 metric, and then mapped back to F then there would be large gains in computational efficiency with respect to other metrics such as those defined on the space of thin shells [15].…”

“…Kilian et al [14] represent surfaces by discrete triangulated meshes and compute geodesic paths (deformations) between them, while assuming that the meshes are registered. Heeren et al [15] proposed a method for computing geodesic-based deformations of thin shell shapes, with extensions for computing summary statistics in the shell space [16], but with known registration. Some other papers solve for registration [4], [17] while ignoring deformation.…”

Differentiable Piecewise-Bézier Surfaces on Riemannian Manifolds

“…The discretized analog, also described in [12], is a discrete shell M h , by which we mean a triangulated surface in R 3 , represented by a tuple (N h , T h ) ∈ (R 3 ) m × ({1, . .…”

“…In the continuous case, a shell S h is given by an oriented C 2 surface S in R 3 , called the midplane of S h , and is defined as the set S h = {p + λν(p) | p ∈ S, λ ∈ − The tangent space at S ∈ S consists of smooth displacement fields ψ : S → R 3 , and it can be equipped with a Riemannian metric that describes the physical energy dissipation during the deformation of S h ; for details we refer to [12].…”