Discrete Differential Geometry

Abstract

The bending energy of a thin, naturally straight, homogeneous and isotropic elastic rod of length L is given byis the arclength parametrisation and κ = γ (s) the curvature vector. Consider the following boundary value problem: Given points P , Q ∈ R m and unit vectors v, w ∈ S m−1 find the shapes of static elastic curves with clamped ends and fixed length. Defining the spacethis can be reformulated to find the minimizers of F : C → R.

Curvature line parametrized surfaces and orthogonal coordinate systems: discretization with Dupin cyclides

“…A circular net x : Z 3 → R N is determined by its values along three coordinate planes intersecting in one point. This is due to the following classical result (see for example [BS08] for the proof).…”

“…Mostly we will omit the variable z and denote the functions values by f, f i , f ij , etc. Circular nets are a discretization of smooth orthogonal nets and objects of Möbius geometry (see [BS08] for details). For N = 3, the case m = 2 is a discretization of curvature line parametrized surfaces in R 3 , where the case m = 3 discretizes triply orthogonal coordinate systems.…”

“…Principal contact element nets are a discretization of curvature line parametrized surfaces in Lie geometry, and they unify the previous discretizations as circular nets in Möbius geometry and as conical nets in Laguerre geometry. For details and further references see [BS08].…”

“…According to the "multidimensional consistency principle" for the discretization of smooth geometries, the transformations of cyclidic nets are defined by the same geometric relations as the nets itself. For a detailed explanation see [BS08,Chap. 3], where Ribaucour transformations for principal contact element nets are introduced in the same spirit.…”

“…Probably the most elaborate source is the classical book [Bla29] by Blaschke, a modern comprehensive introduction is contained in Cecil [Cec92]. A short introduction can be found in [BS08].…”