Evolute

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An ellipse (red) and its evolute (blue), the dots are the vertices of the curve, each vertex corresponds to a cusp on the evolute. the evolute of an ellipse is called an astroid.

How the above evolute is constructed.

In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. Equivalently, it is the envelope of the normals to a curve. The original curve is an involute of its evolute. (Compare Media:Evolute2.gif and Media:Involute.gif)

1 History
2 Equations
3 Properties
- 3.1 Relationship between a curve and its evolute
4 Radial of a curve
5 Examples
6 References

[edit] History

Apollonius (c. 200 BC) discussed evolutes in Book V of his Conics. However, Huygens is sometimes credited with being the first to study them (1673).

[edit] Equations

Let $(x, y) = (x (t), y (t))$ be a parametrically defined plane curve. Let $R = 1 / κ$ be the radius of curvature and $φ$ be the tangential angle. Then the center of curvature at $(x, y)$ is given by $(x - R sinφ, y + R cosφ)$ and we may take $(X, Y) = (x - R sinφ, y + R cosφ)$ as parametric equations for the evolute. We have $(\cos \phi, \sin \phi) = \frac{(x', y')}{(x'^2+y'^2)^{1/2}}$ and $R = 1/\kappa = \frac{(x'^2+y'^2)^{3/2}}{x'y''-x''y'}$ , so we may eliminate $R$ and $φ$ to obtain:

$(X, Y)= (x-y'\frac{x'^2+y'^2}{x'y''-x''y'}, y+x'\frac{x'^2+y'^2}{x'y''-x''y'})$

If the curve (x, y) is parametrized by arc length s (i.e. $(x, y) = (x (s), y (s))$ where $| (x', y') | = 1$ ; see natural parametrization) then this simplifies to: $(X, Y)= (x+\frac{x''}{x''^2+y''^2}, y+\frac{y''}{x''^2+y''^2}).$

[edit] Properties

Differentiating $(X, Y) = (x - R sinφ, y + R cosφ)$ with respect to $s$ we obtain:

$\frac{d}{ds} (X, Y) = (\frac{dx}{ds} - R \cos \phi \frac{d\phi}{ds} - \frac{dR}{ds}\sin \phi, \frac{dy}{ds} - R \sin \phi \frac{d\phi}{ds} + \frac{dR}{ds}\cos \phi)$ .

$\frac{dx}{ds} = \cos \phi$ , $\frac{dy}{ds} = \sin \phi$ and $\frac{d\phi}{ds} = \kappa = 1/R$ , so this simplifies to

$\frac{d}{ds} (X, Y) = (-\frac{dR}{ds}\sin \phi, \frac{dR}{ds}\cos \phi) = \frac{dR}{ds}(-\sin \phi,\cos \phi).$

Which has magnitude $|\frac{dR}{ds}|$ and direction $\phi \pm \pi/2$ . This has the following implications:

The tangential angle of the evolute is $\phi \pm \pi/2$ . (The sign of $\pm \pi/2$ is determined by the sign of $\frac{dR}{ds}$ .)
The tangent to the evolute is normal to the original curve. A curve is the envelope of its tangents so the evolute is also the envelope of the lines normal to the curve.
The arclength along the curve $(X, Y)$ from $(X (s 1), Y (s 1))$ to $(X (s 2), Y (s 2))$ is given by

$\int_{s_1}^{s_2}\frac{dR}{ds} ds = R(s_2)-R(s_1)$ .

The original curve is an involute of the evolute.

If $φ$ can be solved as a function of $R$ , say $φ = g (R)$ , then the Whewell equation for the evolute is $Φ = g (R) + π / 2$ , where $Φ$ is the tangential angle of the evolute and we take $R$ as arclength along the evolute. From this we can derive the Cesàro equation as $Κ = g'(R)$ , where $Κ$ is the curvature of the evolute.

[edit] Relationship between a curve and its evolute

An ellipse (red), its evolute (blue) and some parallel curves. Note how the parallel curves have cusps when they touch the evolute

By the above discussion, the derivative of $(X, Y)$ vanishes when $\frac{dR}{ds} = 0$ , so the evolute will have a cusp when the curve has a vertex, that is when the curvature has a local maximum or minimum. At a point of inflection of the original curve the radius of curvature becomes infinite and so $(X, Y)$ will become infinite, often this will result in the evolute having an asymptote. Similarly, when the original curve has a cusp where the radius of curvature is 0 then the evolute will touch the the original curve.

This can be seen in the figure to the right, the blue curve is the evolute of all the other curves. The cusp in the blue curve corresponds to a vertex in the other curves. The cusps in the green curve are on the evolute. Curves with the same evolute are parallel.

[edit] Radial of a 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 x and y terms from the equation of the evolute. Ths produces $(X, Y) = ( - R sinφ, R cosφ)$ or $(X, Y)= (-y'\frac{x'^2+y'^2}{x'y''-x''y'}, x'\frac{x'^2+y'^2}{x'y''-x''y'}).$