Evolute

The evolute (red) of a curve (parabola, blue) is the geometric location of all centers of curvature or the envelope of their normals

The evolution of a flat curve is

• the path on which the center of the circle of curvature moves when the corresponding point passes through the given curve.

Or:

• the envelope of the normals of the given curve.

Evolutes are closely related to the involutes of a given curve, because the following applies: A curve is the evolute of each of its involutes.

Evolute of a parameterized curve

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

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

the evolution of the given curve.

Is and so is ${\displaystyle {\vec {c}}(t)=(x(t),y(t))^{T}}$ ${\displaystyle {\vec {E}}=(X,Y)^{T}}$

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

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

Properties of the Evolute

Evolute: The normal in P is tangent in M.

In order to derive properties of a regular curve it is advantageous to use the arc length of the given curve as a parameter . Because then applies (see Frenet's formulas ) and . From this follows the evolute for the tangent vector : ${\displaystyle s}$ ${\displaystyle \;|{\vec {c}}'|=1\;}$ ${\displaystyle \;{\vec {n}}'=-{\vec {c}}'/\rho \;}$ ${\displaystyle \;{\vec {E}}={\vec {c}}+\rho {\vec {n}}\;}$

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

The following properties of an Evolute result from this equation:

• The evolute is not regular in points , i.e. That is, it has peaks at points of maximum or minimum curvature (see parabola, ellipse, nephroide). ${\displaystyle \rho '=0}$
• The normals of the given curve are tangents of the evolute, i.e. i.e. the evolute is the envelope of the normals of the given curve.
• In sections of the given curve in which or applies, it is an involute of its evolute. (In the picture the blue parabola is an involute of the red Neil parabola.) ${\displaystyle \rho '>0}$ ${\displaystyle \rho '<0}$

Proof of the last property:
in the section under consideration let . An involute of the evolute can be described as follows: ${\displaystyle \rho '>0}$

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

where a thread extension means (see involute ). With and results ${\displaystyle l_{0}}$
${\displaystyle {\vec {E}}={\vec {c}}+\rho {\vec {n}}\;,\;{\vec {E}}'=\rho '{\vec {n}}}$ ${\displaystyle \rho '>0}$

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

This means that the given curve is obtained again for the thread extension . ${\displaystyle l_{0}=\rho (0)}$

• Parallel curves have the same evolute.

Proof: A to the given curve at a distance parallel curve has the parametric representation and the radius of curvature (see Fig. Parallel curve ) . So the evolution of the parallel curve is ${\displaystyle d}$ ${\displaystyle {\vec {c}}_{d}={\vec {c}}+d{\vec {n}}}$ ${\displaystyle \rho _{d}=\rho -d}$ ${\displaystyle {\vec {E}}_{d}={\vec {c}}_{d}+\rho _{d}{\vec {n}}={\vec {c}}+d{\vec {n}}+(\rho -d){\vec {n}}={\vec {c}}+\rho {\vec {n}}={\vec {E}}\;.}$

Examples

Evolute the normal parabola

The normal parabola can be described by the parametric representation . According to the above formulas, the following equations result for the evolute: ${\displaystyle (t,t^{2})}$

${\displaystyle X=\cdots =-4t^{3}}$

${\displaystyle Y=\cdots ={\frac {1}{2}}+3t^{2}}$

This is the parametric representation of a Neil parabola .

Evolute (red) of an ellipse

The evolution of the large nephroids (blue) is the small nephroids (red)

Evolute of an ellipse

For the ellipse with the parametric representation we get: ${\displaystyle (a\cos t,b\sin t)}$

${\displaystyle X=\cdots ={\frac {a^{2}-b^{2}}{a}}\cos ^{3}t}$

${\displaystyle Y=\cdots ={\frac {b^{2}-a^{2}}{b}}\sin ^{3}t}$

These equations describe a crooked astroid . Elimination of delivers the implicit representation ${\displaystyle t}$

• ${\displaystyle (aX)^{\tfrac {2}{3}}+(bY)^{\tfrac {2}{3}}=(a^{2}-b^{2})^{\tfrac {2}{3}}\ .}$

Evolutions of known curves

• To an astroid : again an astroid (twice as large)
• To an ellipse : a crooked astroid
• To a cardioid : again a cardioid (one third as big)
• To a circle : a point, namely its center
• To a deltoid : again a deltoid (three times as large)
• To a cycloid : a congruent cycloid
• To an epicycloid : an enlarged epicycloid
• To a hypocycloid : a similar hypocycloid
• To a logarithmic spiral : the same logarithmic spiral
• To a nephroid : again a nephroid (half the size)
• To a parable : a Neil parable
• To a tractrix : a catenoid (chain line)

Individual evidence

1. ↑ R.Courant: lectures on differential and integral calculus. Volume 1, Springer-Verlag, 1955, p. 268.

literature

• K. Burg, H. Haf, F. Wille, A. Meister: Vector analysis: Higher mathematics for engineers, natural scientists and ... Springer-Verlag, 2012, ISBN 3-8348-8346-8 , p. 30.
• Small encyclopedia of mathematics. Harry Deutsch Verlag, 1977, ISBN 3-87144-323-9 , p. 475.

Web links

Commons : Evolute  - collection of images, videos and audio files

• Eric W. Weisstein : Evolute . In: MathWorld (English).