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STURM ROULETTE

Curve studied by Sturm in 1841.
Charles-François Sturm (1803 - 1855): French mathematician. |

Differential equation:
with e = 1 for the elliptic
roulette (ellipse with semi-axes Cartesian parametrization in the where Cartesian parametrization in the |

The notion of Sturm roulette
refers to the locus of the centre of a centred conic rolling without slipping
on a line; it is said to be *elliptic* of *hyperbolic* depending
on whether the conic is an ellipse
or a hyperbola.

With a constant major axis, in the elliptic case, the
Sturm roulette goes from being the line (case of the rolling circle e
= 0), to being a reunion of semicircles (case of a "rolling" segment line
e
= 1). |

With a constant major axis, in the hyperbolic case, the
Sturm roulette goes from being a reunion of semicircles (e = 1)
to a segment line (infinite e). |

The special case of a rectangular hyperbola () gives the rectangular Sturm roulette, which also is a lintearia, as well as a Ribaucour curve (cf. right animation at the top of the page).

See also the Delaunay
roulettes, locus of a *focus* of the conic, as well as the determination
of the road associated to an elliptic
wheel.

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© Robert FERRÉOL
2017