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Molding surface with generatrix an epicycloid with 5 cusps and the direction of the parallels of which is a sinusoid.

Surface studied by Monge, and by Pirondini in 1897.
Other name : carved surface.

Parametrization:  where (Mp) is a plane curve (taken in xOy) that is the direction of the parallels, with normal vector  and  is the parametrization of any plane curve (the generatrix).

The area of the portion of surface delimited by two directrices and two generatrices is the product of the length of a portion of generatrix by the length of the curve described by the center of gravity of the portions of generatrices.

A molding surface is a Monge surface the parallels of which are plane curves.

It is therefore by definition a surface that is the union of plane curves parallel to one another.

It is the surface generated by the motion of a curve (a generatrix) on a plane remaining parallel to a fixed curve and all the points of which have a speed vector orthogonal to this plane, in other words, of a plane rolling without slipping on a cylinder.

The curvature lines are the parallels and their orthogonal trajectories (the generatrices).

When the parallels are linear, we get the cylinders (the parallels of them are named the generatrices and the generatrices, the directrices...), and when they are circular, we get the surfaces of revolution (the generatrices are then the meridians).

Example : the movement of a straight line of a  plane rolling without sliding on a cylinder of revolution generates a developable helicoid.
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© Robert FERRÉOL 2020