next curve previous 2D curves 3D curves surfaces fractals polyhedra

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

 
Intrinsic equation: .
Parametrization: , with  ().
Taking , we get:
.

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:
.

 
The case g(t) = t gives the curve:  (cf. Centrale 2 1988 admission exam):

The circular helix (including the circle), is the only curve with constant curvature and torsion.

See also the geodesic circles.
 
 
next curve previous curve 2D curves 3D curves surfaces fractals polyhedra

© Robert FERRÉOL 2018