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In mathematics, the curve-shortening flow is a process that modifies a smooth curve in the Euclidean plane by moving its points perpendicularly to the curve at a speed proportional to the curvature. The curve-shortening flow is an example of a geometric flow, and is the one-dimensional case of the mean curvature flow. Other names for the same process include the Euclidean shortening flow, geometric heat flow, and arc length evolution. As the points of any smooth simple closed curve move in this way, the curve remains simple and smooth. It loses area at a constant rate, and its perimeter decreases as quickly as possible for any continuous curve evolution. If the curve is non-convex, its total absolute curvature decreases monotonically, until it becomes convex. Once convex, the isoperimetric ratio of the curve decreases as the curve converges to a circular shape, before collapsing to a single point of singularity. If two disjoint simple smooth closed curves evolve, they remain disjoint until one of them collapses to a point. The circle is the only simple closed curve that maintains its shape under the curve-shortening flow. However, if curves that cross themselves or have infinite length are allowed, there are other curves that also keep their shape. These include the grim reaper curve, an infinite curve that translates upwards, as well as self-crossing curves that shrink while maintaining their shape and spirals that rotate while remaining the same size and shape. The curve-shortening flow was originally studied as a model for annealing of metal sheets. Later, it was applied in image analysis to give a multi-scale representation of shapes. It can also model reaction–diffusion systems, and the behavior of cellular automata. In pure mathematics, the curve-shortening flow can be used to find closed geodesics on Riemannian manifolds, and as a model for the behavior of higher-dimensional flows. ==Definitions== In the curve-shortening flow, each point of a curve moves in the direction of a normal vector to the curve, at a rate proportional to the curvature. For an evolving curve represented by a two-parameter function where parameterizes the arc length along the curve and parameterizes a time in the evolution of the curve, the curve-shortening flow can be described by the parabolic partial differential equation : a form of the heat equation, where is the curvature and is the unit normal vector. This definition is clearly invariant under translations and rotations of the Euclidean plane. If the plane is scaled by a constant dilation factor, the flow remains essentially unchanged, but is slowed down or sped up by the same factor. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Curve-shortening flow」の詳細全文を読む スポンサード リンク
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