ROULETTE WITH LINEAR BASE

 Notion studied by Besant in 1869.

Given a curve ( ) and a point O linked to ( ), the associated roulette with linear base is the trace of the point O when the curve ( ) rolls without slipping on a fixed line. Therefore, this is a movement of a plane over a fixed plane the base of which is linear.

 The formulas linking the equations of the rolling curve ( ) and of the roulette ( ) are . Starting from a rolling curve , we get the roulette . Conversely, starting from a roulette , we get the rolling curve , the pedal of which can be obtained more simply by the formulas: . The curvilinear abscissa and the radius of curvature of the rolling curve are given by: In complex parametrization, the relation can be written: . Examples:

 figure rolling curve tracing point roulette circle on the circle cycloid circle outside the circle trochoid parabola focus of the parabola catenary centred conic focus of the conic Delaunay roulette centred conic centre of the conic Sturm roulette logarithmic spiral centre of the spiral line involute of a circle centre of the circle parabola the base of which is the symmetry axis Tschirnhausen cubic focus of the cubic parabola the directrix of which is the base of the rolling motion Norwich spiral pole of the spiral Tschirnhausen cubic hyperbolic spiral centre of the spiral tractrix centred cycloid centre of the cycloid ellipse sinusoidal spiral of index n pole of the spiral Ribaucour curve of index 1+1/n

Since the roulettes with linear base of a curve are identical to the glissettes with linear base of any involute of this curve, see other examples at glissette.

Compare to the notion of wheel-road couple, where, now, it is the roulette that is linear instead of the base.