Evolute

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). 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.[3]

If ${\displaystyle {\vec {x}}={\vec {c}}(t),\;t\in [t_{1},t_{2}]}$ is the parametric representation of a regular curve in the plane with its curvature nowhere 0 and ${\displaystyle \rho (t)}$ its curvature radius and ${\displaystyle {\vec {n}}(t)}$ the unit normal pointing to the curvature center, then

• ${\displaystyle {\vec {E}}(t)={\vec {c}}(t)+\rho (t){\vec {n}}(t)}$

describes the evolute of the given curve.

For ${\displaystyle {\vec {c}}(t)=(x(t),y(t))^{T}}$ and ${\displaystyle {\vec {E}}=(X,Y)^{T}}$ one gets

• ${\displaystyle \displaystyle X(t)=x(t)-{\frac {y'(t)\cdot {\Big (}x'(t)^{2}+y'(t)^{2}{\Big )}}{x'(t)\cdot y''(t)-x''(t)\cdot y'(t)}}\quad }$ and

${\displaystyle \displaystyle Y(t)=y(t)+{\frac {x'(t)\cdot {\Big (}x'(t)^{2}+y'(t)^{2}{\Big )}}{x'(t)\cdot y''(t)-x''(t)\cdot y'(t)}}}$.

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 ${\displaystyle s}$ of the given curve as its parameter, because of ${\displaystyle \;|{\vec {c}}'|=1\;}$ and ${\displaystyle \;{\vec {n}}'=-{\vec {c}}'/\rho \;}$ (see Frenet–Serret formulas). Hence the tangent vector of the evolute ${\displaystyle \;{\vec {E}}={\vec {c}}+\rho {\vec {n}}\;}$ is:

${\displaystyle {\vec {E}}'={\vec {c}}'+\rho '{\vec {n}}+\rho {\vec {n}}'=\rho '{\vec {n}}\ .}$

From this equation one gets the following properties of the evolute:

• At points with ${\displaystyle \rho '=0}$ the evolute is not regular. That means: at points with maximal or minimal curvature (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.[4]
• 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 ${\displaystyle \rho '>0}$ or ${\displaystyle \rho '<0}$ 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 ${\displaystyle \rho '>0}$ at the section of consideration. An involute of the evolute can be described as follows:

${\displaystyle {\vec {C}}_{0}={\vec {E}}-{\frac {{\vec {E}}'}{|{\vec {E}}'|}}\;{\Big (}\int _{0}^{s}|{\vec {E}}'(w)|\;\mathrm {d} w+l_{0}\;{\Big )}\;,}$

where ${\displaystyle l_{0}}$ is a fixed string extension (see Involute of a parameterized curve ).
With ${\displaystyle {\vec {E}}={\vec {c}}+\rho {\vec {n}}\;,\;{\vec {E}}'=\rho '{\vec {n}}}$ and ${\displaystyle \rho '>0}$ one gets

${\displaystyle {\vec {C}}_{0}={\vec {c}}+\rho {\vec {n}}-{\vec {n}}\;{\Big (}\int _{0}^{s}\rho '(w)\;\mathrm {d} w\;+l_{0}{\Big )}={\vec {c}}+(\rho (0)-l_{0})\;{\vec {n}}\;.}$

That means: For the string extension ${\displaystyle l_{0}=\rho (0)}$ the given curve is reproduced.

• Parallel curves have the same evolute.

Proof: A parallel curve with distance ${\displaystyle d}$ off the given curve has the parametric representation ${\displaystyle {\vec {c}}_{d}={\vec {c}}+d{\vec {n}}}$ and the radius of curvature ${\displaystyle \rho _{d}=\rho -d}$ (see parallel curve). Hence the evolute of the parallel curve is ${\displaystyle {\vec {E}}_{d}={\vec {c}}_{d}+\rho _{d}{\vec {n}}={\vec {c}}+d{\vec {n}}+(\rho -d){\vec {n}}={\vec {c}}+\rho {\vec {n}}={\vec {E}}\;.}$

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 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.

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. This produces

${\displaystyle (X,Y)=\left(-y'{\frac {{x'}^{2}+{y'}^{2}}{x'y''-x''y'}}\;,\;x'{\frac {{x'}^{2}+{y'}^{2}}{x'y''-x''y'}}\right)\;.}$

• Weisstein, Eric W. "Evolute". MathWorld.
• Sokolov, D.D. (2001) [1994], "Evolute", Encyclopedia of Mathematics, EMS Press
• Yates, R. C.: A Handbook on Curves and Their Properties, J. W. Edwards (1952), "Evolutes." pp. 86ff
• Evolute on 2d curves.