The locus of the center of curvatureMathworldPlanetmath of a plane curveMathworldPlanetmath is called the evoluteMathworldPlanetmath of this curve.

The coordinates of the center of curvature belonging to the point  P=(x,y)  of the curve γ are

ξ=x-ϱsinα,η=y+ϱcosα, (1)

where ϱ is the radius of curvature in P and α is the slope angle of the tangent lineMathworldPlanetmath of the curve in P.  So (1) may be regarded as the equations of the evolute of γ.

If the plane curve is given in the parametric form   x=x(t),y=y(t),  the corresponding parametric equations of the evolute are

ξ=x-(x 2+y 2)yxy′′-x′′y,η=y+(x 2+y 2)xxy′′-x′′y.

In the spexial case that the curve is given in the form  y=y(x)  these equations can be written

ξ=x-(1+y 2)yy′′,η=y+1+y 2y′′.

For examining the properties of the evolute we choose for parameter the arc lengthMathworldPlanetmath s, measured from a certain point of the curve; then in (1) the quantities x,y,ϱ,α and thus ξ and η are functions of s. We assume that all needed derivatives exist and are continuousMathworldPlanetmath.

Differentiating (1) with respect to s, we obtain


and recalling that  dxds=cosα,  dyds=sinα  and  ϱdαds=1  it yields

dξds=-dϱdssinα,dηds=dϱdscosα. (2)

If  dϱds0  in the point  (x,y)  of γ, the derivatives dξds and dηds do not vanish simultaneously, and so the evolute has in the corresponding point  (ξ,η)  a tangent line with the slope


Since the of this is the slope of the normal lineMathworldPlanetmath of the given curve γ, we have the

Theorem 1.  The normal line of the curve in a point  (x,y),  where  dϱds0, is the tangent line of the evolute, having as tangency point the corresponding center of curvature  (ξ,η).  Thus the evolute is the envelopeMathworldPlanetmath of the normal lines of the curve.

We shall calculate the arc length σ of the evolute corresponding the arc of the curve γ which is passed through when the parameter s grows from s1 to s2; we assume that ϱ and dϱds are then continuous and distinct from zero. According the arc length formula,


Using the equations (2) and the fact that the sign of dϱds does not change, we can write


where ϱ1 and ϱ2 are the corresponding of γ. We have proved the

Theorem 2.  The of an arc of the evolute is equal to the difference of the of the given curve touching the arc of the evolute in its end pointsPlanetmathPlanetmath, provided that ϱ and dϱds are continuous and do not change their sign on the arc of the curve.


  • 1 Ernst Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset I. Toinen painos. WSOY, Helsinki (1950).
Title evolute
Canonical name Evolute
Date of creation 2013-03-22 17:35:06
Last modified on 2013-03-22 17:35:06
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Topic
Classification msc 53A04
Related topic ConditionOfOrthogonality
Related topic ArcLength
Related topic BolzanosTheorem
Related topic SubstitutionNotation
Defines evolute