On the topology of the spaces of curvature constrained plane curves

  • 1 Instituto de Ciencias Exactas y Naturales, Universidad Arturo Prat, Av. Arturo Prat 2120, Iquique, Chile
José Ayala
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  • Instituto de Ciencias Exactas y Naturales, Universidad Arturo Prat, Av. Arturo Prat 2120, Iquique, Chile
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Plane curves with the same endpoints are homotopic; an analogous claim for plane curves with the same endpoints and bounded curvature still remains open. We find necessary and sufficient conditions for two plane curves with bounded curvature to be deformed into each other by a continuous one-parameter family of curves also having bounded curvature. We conclude that the space of these curves has either one or two connected components, depending on the distance between the endpoints. The classification theorem presented here answers a question raised in 1961 by L. E. Dubins.

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Advances in Geometry is a mathematical journal which publishes original research articles of excellent quality in the area of geometry. Geometry is a field of long-standing tradition and eminent importance. The study of space and spatial patterns is a major mathematical activity, and geometric ideas and the geometric language permeate all of mathematics.