Evolute

The evolute of a plane curve is the path on which moves the center of the circle of curvature when the contact point moves along the curve along. Or: The evolute of a curve is the envelope or envelope of its normals.

At the opposite figure: output curve is shown in black normal parabola. K, K1, K2, ... have their curvature circles at the points S, P1, P2, ..., whose center points M, M1, M2, ... constituting the evolute for the normal parabola. The right-hand branch of the evolute arises when the parabola points P migrate to the left.

For a plane curve with the parametric representation

Is the parametric representation of the evolutes given by the coordinates of the center of curvature

And

The output curve from which an evolute arises is, ( with regard to the evolute ) whose involute or its involute.

Evolutes known curves

• To a astroid: turn a astroid ( twice as large )
• For an ellipse: an oblique astroid
• To a cardioid: turn a cardioid (one -third the size )
• For a circle: a point, namely the center of which
• To a deltoids: turn a deltoids ( three times as large )
• For a cycloid: a congruent cycloid
• To a Epicycloid an enlarged Epicycloid
• To a hypocycloid is a similar hypocycloid
• To a logarithmic spiral: The same logarithmic spiral
• To a Nephroide: turn a Nephroide ( half the size )
• To a parable: a parable Neilsche
• To a tractrix: a catenoid ( chain line)

All is equidistant from the output curve have the same evolute.