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INVOLUTE OF A CURVE, OF A DEVELOPABLE SURFACE
Notion studied by Monge in 1771. 
If M_{0} is the current point on (G_{0}), the current point M of an involute of this curve is the locus of the points ;
Cartesian parametrization: . Radius of curvature: . 
I) Involutes of a curve.
The involutes of a curve (G_{0}) are the trajectories described by the points of a line rolling without slipping on this curve.
Therefore, they trace, on the tangent developable (S_{0}) of (G_{0}), the orthogonal trajectories to these tangents; they also form an equivalence class of parallel curves.
They are also the curves for which the initial curve is an evolute.
When the surface (S_{0}) is developed on a plane, the 3D involutes become the 2D involutes of the image of (G_{0}) on this plane.
An involute is planar iff the initial curve is a helix. In this case, all the involutes are planar.
II) Involutes of a developable surface.
The involutes of a developable surface (S_{0}) are the trajectories of a point on a plane rolling without slipping on the surface (S_{0}); therefore, they are the curves for which the initial surface is the polar developable. They present a cuspidal point on (S_{0}), with a tangent that is orthogonal to (S_{0}); and they form an equivalence class of parallel curves.
When (S_{0}) is a cylinder, the involutes are the involutes of the planar sections perpendicular to the axis.
When (S_{0}) is a cone, the involutes are traced on spheres centered on the vertex of the cone: they are the loci of a point on a circle centered on the vertex of the cone rolling without slipping on an orthogonal trajectory of the generatrices of the cone.
In the case of a cone of revolution, they are spherical helices. 

Problem: how to derive the parametrization of the involutes of (S_{0})
from the parametrization of its cuspidal edge
????
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© Robert FERRÉOL 2018