Sturm Curves

Transcendental curves in the plane, described by a point associated with an ellipse, hyperbola or parabola, as it rolls along a straight line. An example of a Sturm curve is the trajectory of the focus of a parabola as it rolls along the $x$-axis — a catenary.

These curves were studied by J.Ch. Sturm.

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Replacing "rolls along a straight line" by "rolls along another fixed curve" , the point will describe a roulette.

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