The evolution of space curves by curvature and torsion

Abstract

We apply Lie group based similarity methods to the study of a new, and widely relevant, class of objects, namely motions of a space curve. In particular, we consider the motion of a curve evolving with a curvature κ and torsion τ dependent velocity law. We systematically derive the Lie point symmetries of all such laws of motion and use these to catalogue all their possible similarity reductions. This calculation reveals special classes of law with high degrees of symmetry (and a correspondingly large number of similarity reductions). Of particular note is one class which is invariant under general linear transformations in space. This has potential applications in pattern and signal recognition.

Curvatures, volumes and norms of derivatives for curves in Riemannian manifolds

“…Comparing this with our previous expression for < (e 1 × · · · × e r ) ′ , e r ∧ e r+2 ∧ · · · ∧ e n >, we obtain equation (17). Let now r = n − 1.…”

“…In this section, M is a non-orientable riemannian manifold. 17 The In conclusion we note that there are non-orientable two-point homogeneous spaces, e. g., the even-dimensional real projective spaces.…”

“…, κ n−1 (x, t) be its curvatures. Concluding this section, we note that evolutions of curves in R 3 by the curvature and torsion have applications to turbulence and to DNA analysis [17]. 3.…”

“…Since these identities are equivalent to equations (17) and (18) respectively, our Theorem 1 is not new. However, equations (17) and (18) are more direct than equations (28) and (29); our derivation of these identities is elementary and straightforward. 11 For these reasons we feel that Theorem 1 deserves publication.…”

Minimal surface generating flow for space curves of non-vanishing torsion

“…The motion law derived in this text involves the torsion of the curve. Geometric flows with torsion were previously considered in [36]. Existence of the solution of related geometric flows was studied in e.g.…”