Similarity solutions in the theory of curvature driven diffusion along planar curves: II. Curves that travel at constant speed

In the theory of curvature driven diffusion along curves, the rate upsilon at which a planar curve C = C(t) advances along its normal vector is propor tional to the second derivative of the curvature ii with respect to the cur ve's are-length parameter, s, i.e., upsilon(s, t) = Ak(ss),(S, t). The curv e is called invariant if it evolves without deformation or rotation; its mo tion is then a steady translation, and the angle theta = theta (s) from the direction of propagation of C to the tangent vector at s obeys the equatio n A theta"'(s)(s) = V sin theta(s) in which V is the speed of propagation. When C is an infinite curve, this equation with V > O implies that as s --> +infinity or -infinity, C either is asymptotic to a straight line parallel to the direction of propagation or spirals to a limit point with YI(S) app roaching a non-zero constant. If C spirals to a point x(+infinity) as s inc reases to +infinity, C may either spiral to a point x(-infinity) or be asym ptotic to a line l(-) as s decreases to -infinity, The curves that are asym ptotic to lines both as s --> +infinity and as s --> -infinity differ by on ly similarity transformations and are such that El = l(-) and have that lin e as an axis of symmetry. A discussion is given of properties that data of the form (theta(0), theta'(0), theta"(0)) must have to determine a curve as ymptotic to a line fur either large or small s, (C) 1999 Elsevier Science B .V, All rights reserved.