Evolute of a curve

EVOLUTE OF A CURVE

Notion studied by Monge in 1771.
Other names: focal or caustic curve.

Given an initial curve with current point , an evolute is a set of points where the angle is the torsion angle, defined up to a constant (see the notations).

An evolute of a curve is a curve the tangents of which are the normals of the initial curve. In other words, it is an envelope of a normal to the initial curve. The evolutes are traced on the polar developable of the initial curve, and they are geodesics (curves that develop onto straight lines). Therefore, we obtain an evolute by uncoiling a wire from a point of the curve to the polar surface (the wire being tangent to it) and recoiling it around the surface.

When a normal plane describes the initial curve, it spins without slipping on the polar surface and, at the same time, the normal that envelopes one of the evolutes rolls without slipping on this evolute: the evolutes are the curves for which the initial curve is an involute.

Evolutes have a cuspidal point on the cuspidal edge of the polar surface. This edge is their envelope.

Note: the locus of the centers of curvature of a non planar curve is not an evolute of the curve; the principal normals do not have an envelope. However, the projection on the osculating plane of the characteristic point of a normal that envelopes an evolute is the center of curvature.

A planar curve (C) has one planar evolute and infinitely many 3D evolutes that are cuspidal edges of the surfaces of constant slope with directrix (C) that can be projected onto the planar evolute.

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