HistoryApollonius of Perga, Apollonius ( 200 BC) discussed evolutes in Book V of his ''Conics''. However, Christiaan Huygens, Huygens is sometimes credited with being the first to study them (1673). Huygens formulated his theory of evolutes sometime around 1659 to help solve the problem of finding the tautochrone curve, which in turn helped him construct an isochronous pendulum. This was because the tautochrone curve is a cycloid, and the cycloid has the unique property that its evolute is also a cycloid. The theory of evolutes, in fact, allowed Huygens to achieve many results that would later be found using calculus.
Evolute of a parametric curveIf is the parametric representation of a regular curve in the plane with its curvature nowhere 0 and its curvature radius and the unit normal pointing to the curvature center, then * describes the evolute of the given curve. For and one gets * and : .
Properties of the evoluteIn order to derive properties of a regular curve it is advantageous to use the arc length of the given curve as its parameter, because of and (see Frenet–Serret formulas). Hence the tangent vector of the evolute is: : From this equation one gets the following properties of the evolute: *At points with the evolute is ''not regular''. That means: at points with maximal or minimal curvature (vertex (curve), vertices of the given curve) the evolute has ''cusps'' (s. parabola, ellipse, nephroid). *For any arc of the evolute 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:
Evolute of a parabolaFor the parabola with the parametric representation one gets from the formulae above the equations: : : which describes a semicubic parabola
Evolute of an ellipseFor 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 cycloidFor the cycloid with the parametric representation the evolute will be: : : which describes a transposed replica of itself.
Evolutes of some curvesThe evolute * of a parabola is a semicubic parabola (see above), * of an ellipse is a non symmetric astroid (see above), *of a Line (geometry), 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 curve, 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 curveA curve with a similar definition is the radial of a given curve. For each point on the curve take the vector from the point to the center of curvature and translate it so that it begins at the origin. Then the locus of points at the end of such vectors is called the radial of the curve. The equation for the radial is obtained by removing the and terms from the equation of the evolute. This produces :
References* * * Yates, R. C.: ''A Handbook on Curves and Their Properties'', J. W. Edwards (1952), "Evolutes." pp. 86ff