Support function at inflection points of planar curves

Abstract We study the support function in the neighborhood of inflections of oriented planar curves. Even for a regular curve, the support function is not regular at the inflection and is multivalued on its neighborhood. We describe this function using an implicit algebraic equation and the rational Puiseux series of its branches. Based on these results we are able to approximate the curve at its inflection to any desired degree by curves with a simple support function, which consequently possess rational offsets. We also study the G 1 Hermite interpolation at two points of a planar curve. It is reduced to the functional C 1 interpolation of the support function. For the sake of comparison and better understanding, we show (using standard methods) that its approximation order is 4 for inflection-free curves. In the presence of inflection points this approximation is known to be less efficient. We analyze this phenomenon in detail and prove that by applying a nonuniform subdivision scheme it is possible to receive the best possible approximation order 4, even in the inflection case.

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