INVOLUTE OF A CURVE, OF A DEVELOPABLE SURFACE

 Notion studied by Monge in 1771. Involute of a conical spiral

 If M0 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  ????