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INVOLUTE OF A CURVE, OF A DEVELOPABLE SURFACE
Notion studied by Monge in 1771. 
Involute of a conical spiral
If M_{0} is the current
point on ,
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 are the trajectories described by the points of a line rolling without slipping on this curve.
Therefore, they trace, on the tangent
developable
of , 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 is developed on a plane, the 3D involutes become the 2D involutes of the image of 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 are the trajectories of a point on a plane rolling without slipping on the surface ; therefore, they are the curves for which the initial surface is the polar developable. They present a cuspidal point on , with a tangent that is orthogonal to ; and they form an equivalence class of parallel curves.
When is a cylinder, the involutes are the involutes of the planar sections perpendicular to the axis.
When
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 from
the parametrization of its cuspidal edge ????
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© Robert FERRÉOL 2018