next curve  previous curve  2D curves  3D curves  surfaces  fractals  polyhedra 
DELAUNAY ROULETTE
Curve studied by Delaunay, 1841; Lindelöf, 1861.
CharlesEugène Delaunay (1816  1872): French astronomer. Other names: elliptic, parabolic, hyperbolic catenary. 
Differential equation: with e = 1 for the elliptic
roulette (ellipse with semiaxes a and b (a > b)),
Length on a period: .

The Delaunay roulette
is the locus of one of the foci of a conic rolling without slipping on
a line. It is elliptic, parabolic, or hyperbolic,
depending on whether the conic is an ellipse,
a parabola, or a hyperbola.


The parabolic roulette of Delaunay is none other than the catenary. 
With Ox as the rolling axis, the Delaunay roulette is a curve oscillating between the lines and . The line y = b is where can be found: in the elliptic case, inflection points with a slope equal to ; and points with vertical tangent in the hyperbolic case. The period of y as a function of x is given by the elliptic integral l(2p); it is the length of the ellipse in the elliptic case.
The following ingenious articulated mechanism can trace
Delaunay roulettes:




This animation shows that, in the parabolic case, the directrix of the parabola envelopes the symmetric catenary about the rolling axis. This directrix cuts the rolling line at the same point than the tangent to the roulette at the corresponding point. 
These curves were considered by Delaunay because of their property of being the only meridians of the surfaces of revolution with constant mean curvature, which are the Delaunay surfaces.
See also the Sturm roulettes, locus of the centre of the conic, as well as the determination of the road associated to an elliptic wheel.
Link: www.gang.umass.edu/gallery/cmc/cmcgallery0101.html
next curve  previous curve  2D curves  3D curves  surfaces  fractals  polyhedra 
© Robert FERRÉOL 2017