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SKEW CIRCLE
Curve studied by Monge
in 1771, then by Cesaro in 1896 [geometria intrinseca, p.144]
who named it.
Other names: curve with constant curvature, Monge curve. See [Loria 3d] p. 99 and Centrale 2 1988 admission exam. 
Intrinsic equation: .
Parametrization: , with (). Taking , we get: with . Opposite, the case g(t) = t, with an animation of the osculating sphere. 

A skew circle is a skew curve with constant curvature; a necessary and sufficient condition, in the non planar case (nonzero torsion), is that the radius of curvature be equal to the radius of the osculating sphere, or that the osculating sphere be centered on the center of curvature (see the notations).
We can also consider the curves with constant torsion:
Intrinsic equation: .
Parametrization: , with (). Taking , we get: with . In this case, the radius of curvature is : . 
The case gives the curve: . 

The circular helix (including the circle), is the only curve with constant curvature and torsion.
These curves are particular cases of Bertrand curves.
See also the geodesic
circles.
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© Robert FERRÉOL 2018