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[[File:Evolute-parab-1-e.svg|thumb|The '''evolute''' of a [[curve]] (blue parabola) is the locus of all its centers of curvature (red).]]
[[Image:Evolute1.gif|right|thumb|The evolute of a curve (in this case, an ellipse) is the envelope of its normals.]]
In the [[differential geometry of curves]], the '''evolute''' of a [[curve]] is the [[locus (mathematics)|locus]] of all its [[Osculating circle|centers of curvature]]. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that curve. The evolute of a circle is therefore a single point at its center.<ref>{{MathWorld|title=Circle Evolute|id=CircleEvolute|accessdate=5 December 2012}}</ref>  Equivalently, an evolute is the [[envelope (mathematics)|envelope]] of the [[perpendicular|normals]] to a curve.

The evolute of a curve, a surface, or more generally a [[submanifold]], is the [[caustic (mathematics)|caustic]] of the normal map. Let {{mvar|M}} be a smooth, regular submanifold in {{math|ℝ<sup>''n''</sup>}}. For each point {{mvar|p}} in {{mvar|M}} and each vector {{math|'''v'''}}, based at {{mvar|p}} and normal to {{mvar|M}}, we associate the point {{math|''p'' + '''v'''}}. This defines a [[Lagrangian map]], called the normal map. The caustic of the normal map is the evolute of {{mvar|M}}.<ref name="Arnold">{{Cite book|first=V. I.|last=Arnold|first2=A. N.|last2=Varchenko|first3=S. M.|last3=Gusein-Zade|title=The Classification of Critical Points, Caustics and Wave Fronts: Singularities of Differentiable Maps, Vol 1|publisher=[[Birkhäuser]]|year=1985|isbn=0-8176-3187-9}}</ref>

Evolutes are closely connected to [[involute]]s: A curve is the evolute of any of its involutes.

== History ==
[[Apollonius of Perga|Apollonius]] ({{circa}} 200&nbsp;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.<ref>{{cite book|last=Yoder|first=Joella G.|title=Unrolling Time: Christiaan Huygens and the Mathematization of Nature|publisher=[[Cambridge University Press]]|date=2004}}</ref>

== Evolute of a parametric curve ==
If <math>\vec x= \vec c(t),\; t\in [t_1,t_2]</math> is the parametric representation of a [[regular curve]] in the plane with its curvature nowhere 0 and <math>\rho(t)</math> its curvature radius and <math>\vec n(t)</math> the unit normal pointing to the curvature center, then
* <math>\vec E(t)=\vec c(t) +\rho (t)\vec n(t)</math>
describes the  '''evolute''' of the given curve.

For <math>\vec c(t)=(x(t),y(t))^T</math> and <math>\vec E=(X,Y)^T</math> one gets
* <math>\displaystyle X(t) = x(t) - \frac{y'(t) \cdot \Big(x'(t)^2+y'(t)^2\Big)}{x'(t) \cdot y''(t) - x''(t) \cdot y'(t)}\quad</math> and
: <math>\displaystyle Y(t) = y(t) + \frac{x'(t) \cdot \Big(x'(t)^2+y'(t)^2\Big)}{x'(t) \cdot y''(t) - x''(t) \cdot y'(t)}</math>.

== Properties of the evolute ==
[[File:Evolute-parab-2-e.svg|thumb|The normal at point P is the tangent at the curvature center C.]]
In order to derive properties of a regular curve it is advantageous to use the [[arc length]] <math>s</math> of the given curve as  its parameter, because of <math>\;|\vec c'|=1\;</math> and <math>\;\vec n'=-\vec c'/\rho\;</math> (see [[Frenet–Serret formulas]]). Hence the tangent vector of the evolute <math>\;\vec E=\vec c +\rho \vec n \;</math> is:
:<math>\vec E'=\vec c' +\rho'\vec n +\rho\vec n'=\rho'\vec n\ .</math>
From this equation one gets the following properties of the evolute:
*At points with <math>\rho'=0</math> the evolute is ''not regular''. That means: at points with maximal or minimal curvature the evolute has ''cusps'' (s. parabola, ellipse, nephroid).
*The normals of the given curve are tangents to the evolute. Hence: the evolute is the ''envelope of the normals'' of the given curve. 
*At sections of the curve with <math>\rho'>0</math> or <math>\rho'<0</math> 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:<br />
Let be <math>\rho'>0</math> at the section of consideration. An [[involute]] of the evolute can be described as follows:
:<math>\vec C_0=\vec E -\frac{\vec E'}{|\vec E'|}\;\Big(\int_0^s|\vec E'(w)|\; \mathrm dw+l_0\;\Big)\; ,</math>
where <math>l_0</math> is a fixed string extension (see [[involute#Involute of a parameterized curve|Involute of a parameterized curve]] ).<br />
With <math>\vec E=\vec c +\rho\vec n\; ,\; \vec E'=\rho'\vec n</math> and <math>\rho'>0</math> one gets 
:<math>\vec C_0 = \vec c +\rho\vec n-\vec n \;\Big(\int_0^s\rho'(w)\; \mathrm dw\;+l_0\Big)= \vec c +(\rho(0)-l_0)\; \vec n\; .</math>
That means: For the string extension <math>l_0=\rho(0)</math> the given curve is reproduced.

*''Parallel curves'' have the same evolute.
''Proof:'' A parallel curve with distance <math>d</math> off the given curve has the parametric representation <math>\vec c_d =\vec c +d\vec n</math> and the radius of curvature <math>\rho_d=\rho -d</math> (see [[parallel curve#Parallel curve of a parametricaly given curve|parallel curve]]). Hence the evolute of the parallel curve is <math>\vec E_d=\vec c_d +\rho_d \vec n=\vec c +d\vec n +(\rho -d)\vec n=\vec c +\rho \vec n = \vec E\; .</math>

== Examples ==
=== Evolute of a parabola ===
For the parabola with the parametric representation <math>(t,t^2)</math> one gets from the formulae above the equations:
:<math>X=\cdots=-4t^3</math>
:<math>Y=\cdots=\frac{1}{2}+3t^2 \; ,</math>
which describes a [[semicubic parabola]]

[[File:Evolute-elli.svg|thumb|Evolute (red) of an ellipse]]
=== Evolute of an ellipse ===
For the ellipse with the parametric representation <math>(a\cos t, b\sin t)</math> one gets:<ref>R.Courant: ''Vorlesungen über Differential- und Integralrechnung.'' Band 1, Springer-Verlag, 1955, S. 268.</ref>
:<math>X=\cdots = \frac{a^2-b^2}{a}\cos ^3t</math>
:<math>Y=\cdots = \frac{b^2-a^2}{b}\sin ^3t \; .</math>
These are the equations of a non symmetric [[astroid]]. 
Eliminating parameter <math>t</math> leads to the implicit representation
* <math>(aX)^{\tfrac{2}{3}} +(bY)^{\tfrac{2}{3}}=(a^2-b^2)^{\tfrac{2}{3}}\ .</math>

[[File:Cycloid osculating circle evolute 2.gif|300px|thumb|Cycloid (blue), its osculating circle (red) and evolute (green).]]

=== Evolute of a cycloid===
For the [[cycloid]] with the parametric representation <math>(r(t - \sin t), r(1 - \cos t))</math> the evolute will be:<ref>{{MathWorld |title=Cycloid Evolute |id=CycloidEvolute}}</ref>
:<math>X=\cdots=r(t + \sin t)</math>
:<math>Y=\cdots=r(\cos t - 1)</math>
which describes a transposed replica of itself.
[[File:Nephroide-evolute-e.svg|300px|thumb|The evolute of the large nephroid (blue) is the small nephroid (red).]]
== 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 [[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 curve ==
A 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 {{mvar|x}} and {{mvar|y}} terms from the equation of the evolute. This produces

:<math>(X, Y)= \left(-y'\frac{{x'}^2+{y'}^2}{x'y''-x''y'}\; ,\; x'\frac{{x'}^2+{y'}^2}{x'y''-x''y'}\right)\; .</math>

==References==

{{reflist}}
* {{MathWorld|title=Evolute|urlname=Evolute}}
* {{springer|title=Evolute|id=E/e036670|last=Sokolov|first=D.D.}}
* Yates, R. C.: ''A Handbook on Curves and Their Properties'', J. W. Edwards (1952), "Evolutes." pp.&nbsp;86ff
* [http://www.2dcurves.com/derived/curvature.html#evolute Evolute on 2d curves.]

{{Differential transforms of plane curves}}
{{Christiaan Huygens}}

[[Category:Differential geometry]]
[[Category:Curves]]
[[Category:Christiaan Huygens]]