| Popis: |
In the theory of curvature driven diffusion along curves, the rate υ at which a planar curve C=C(t) advances along its normal vector is proportional to the second derivative of the curvature κ with respect to the curve’s arc-length parameter, s, i.e., υ(s,t)=Aκss(s,t). The curve is called invariant if it evolves without deformation or rotation; its motion is then a steady translation, and the angle θ=θ(s) from the direction of propagation of C to the tangent vector at s obeys the equation Aθ″′(s)=V sin θ(s) in which V is the speed of propagation. When C is an infinite curve, this equation with V>0 implies that as s→+∞ or −∞,C either is asymptotic to a straight line parallel to the direction of propagation or spirals to a limit point with κ′(s) approaching a non-zero constant. If C spirals to a point x+∞ as s increases to +∞,C may either spiral to a point x−∞ or be asymptotic to a line l − as s decreases to −∞. The curves that are asymptotic to lines both as s→+∞ and as s→−∞ differ by only similarity transformations and are such that l + = l − and have that line as an axis of symmetry. A discussion is given of properties that data of the form (θ(0),θ′(0),θ″(0)) must have to determine a curve asymptotic to a line for either large or small s. |
Full Text from ScienceDirect