A distance on curves modulo rigid transformations

Abstract

We propose a geometric method for quantifying the difference between parametrized curves in Euclidean space by introducing a distance function on the space of parametrized curves up to rigid transformations (rotations and translations). Given two curves, the distance between them is defined as the infimum of an energy functional which, roughly speaking, measures the extent to which the jet field of the first curve needs to be rotated to match up with the jet field of the second curve. We show that this energy functional attains a global minimum on the appropriate function space, and we derive a set of first-order ODEs for the minimizer.MSC classification: 58E30 (Primary), 49Q10, 53A04 (Secondary).

Importance and Effectiveness of Representing the Shapes of Cosserat Rods and Framed Curves as Paths in the Special Euclidean Algebra

“…Evidently, there is a considerable simplification in the model, since only three scalar fields determine the shape of a rod constrained in accord with (8) and (9). Perhaps surprisingly, however, there is absolutely no simplification in the Lie algebra and group necessary to describe the system.…”

“…Significantly, the general variational approach devised by Schuricht [4] to study the equilibria of nonlinearly elastic rods with topological constraints (and recently adopted by Giusteri, Lussardi & Fried [5] to study the Kirchhoff-Plateau problem) is tacitly based on the same Lie algebraic representation of the rod shape. Moreover, the role of the special Euclidean algebra is also essential in connection with the geometric mechanical concepts described, for instance, in the works by Simo, Marsden & Krishnaprasad [6], Simo, Posbergh & Marsden [7], Holm, Noakes & Vankerschaver [8], and Eldering & Vankerschaver [9] and with the G-strand equations discussed by Holm & Ivanov [10]. It should be noted, however, that these authors apply geometric concepts to study the dynamics of rods, whereas we focus our attention on the description of shapes.…”

“…Moreover, the role of the special Euclidean algebra is also essential in connection with the geometric mechanical concepts described, for instance, in the works by Simo, Marsden and Krishnaprasad [6], Simo, Posbergh and Marsden [7], Holm, Noakes and Vankerschaver [8], and Eldering and Vankerschaver [9] and with the G-strand equations discussed by Holm and Ivanov [10]. It should be noted, however, that these authors apply geometric concepts to study the dynamics of rods, whereas we focus on the description of shapes.…”