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PSEUDOCYCLOIDAL CURVE, PARACYCLOID, HYPERCYCLOID
Curves studied by Euler in 1750 and 1783, Cesaro in 1887. 
The pseudocycloidal curves are the curves satisfying an intrinsic equation of the type , by analogy with the cycloidal curves, which satisfy: . In other words, they are the curves for which, when they roll without slipping on a line, the centre of curvature describes a hyperbola an axis of which is parallel to the line (see Mannheim curve).
There are two cases, depending on the sign of the above constant cte.
First case, cte <0: PARACYCLOID
Cartesian parametrization: .
Cartesian tangential angle: . Curvilinear abscissa: . Radius of curvature: . Intrinsic equation 1: . 

Second case, cte >0: HYPERCYCLOID (be careful, the word
hypercycloid is also used instead of epicycloid)
Cartesian parametrization: .
Cartesian tangential angle: . Curvilinear abscissa: . Radius of curvature: . Intrinsic equation 1: . 

REMARK: the paracycloid is the negative
pedal of the spiral of the hyperbolic sine ,
and the hypercycloid is the negative pedal of the spiral of the hyperbolic
cosine .
Furthermore, they are the evolutes
of one another.
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© Robert FERRÉOL 2017