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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 a and b (a > b)), 
e = 1 for the hyperbolic roulette (hyperbola with semi-axes a and b).
Cartesian equation: .

Cartesian parametrization in the elliptic case: 

where, e = c / a.
Curvilinear abscissa: .

Cartesian parametrization in the hyperbolic case: .

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