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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 

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 wheelroad couple, where, now, it is the roulette that is linear instead of the base.
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© Robert FERRÉOL 2017