that does not include a cusp, the length of the arc equals the difference between the radii of curvature at its endpoints. This fact leads to an easy proof of the Tait–Kneser theorem on nesting of osculating circles. The normals of the given curve at points of nonzero curvature are tangents to the evolute, and the normals of the curve at points of zero curvature are asymptotes to the evolute. Hence: the evolute is the envelope of the normals of the given curve. At sections of the curve with or the curve is an involute of its evolute. (In the diagram: The blue parabola is an involute of the red semicubic parabola, which is actually the evolute of the blue parabola.) Proof of the last property: Let be at the section of consideration. An involute of the evolute can be described as follows: where is a fixed string extension (see Involute of a parameterized curve ). With and one gets That means: For the string extension the given curve is reproduced. Parallel curves have the same evolute. Proof: A parallel curve with distance off the given curve has the parametric representation and the radius of curvature (see parallel curve). Hence the evolute of the parallel curve is Examples Evolute of a parabola For the parabola with the parametric representation one gets from the formulae above the equations: which describes a semicubic parabola Evolute of an ellipse For the ellipse with the parametric representation one gets: These are the equations of a non symmetric astroid. Eliminating parameter leads to the implicit representation Evolute of a cycloid For the cycloid with the parametric representation the evolute will be: which describes a transposed replica of itself. Evolutes of some curves The evolute of a parabola is a semicubic parabola (see above), of an ellipse is a non symmetric astroid (see above), of a line is an ideal point, of a nephroid is a nephroid (half as large, see diagram), of an astroid is an astroid (twice as large), of a cardioid is a cardioid (one third as large), of a circle is its center, of a deltoid is a deltoid (three times as large), of a cycloid is a congruent cycloid, of a logarithmic spiral is the same logarithmic spiral, of a tractrix is a catenary. Radial curve A curve with a similar definition is the radial of a given curve. For each