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CURVE OF CONSTANT PRECESSION
Curve studied by Paul D. Scofield in 1995. 
Cartesian parametrization: where .
Curve traced on the hyperboloid: .

The curves of constant precession are the curves such that the instant rotation vector of the Frenet frame has a motion of uniform rotation around a fixed axis when this frame travels along the curve at constant speed. Therefore, this vector has a motion similar to that of the axis of a spinning top, hence the expression "constant precession".
Remember that if the Frenet frame is written , then the instant rotation vector is defined by the relations: and given by the formula: .
The projection on the plane xOy of the curve given above is an epitrochoid with parameter .
Opposite, the case k =3/5 which gives q = 3.


The end of the tangent vector describes the spherical indicatrix of curvature of the curve; the formulas
show that this indicatrix is a spherical helix. Opposite, the indicatrix of curvature of the curve above. 

Compare to the Capareda curves.
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© Robert FERRÉOL 2018