Curve of constant precession

CURVE OF CONSTANT PRECESSION

Curve studied by Paul D. Scofield in 1995.

Cartesian parametrization:

where

Curve traced on the hyperboloid: .
Curvilinear abscissa: .
Radius of curvature: , radius of torsion: .
Intrinsic equation: .
Instant rotation vector: .

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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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.