Editing Involute (section)

== Properties of involutes ==
[[File:Involute(in red) of parabola(dark blue).png|alt=|thumb|Involute: properties. The angles depicted are 90 degrees.]]
In order to derive properties of a regular curve it is advantageous to suppose the [[arc length]] <math>s</math> to be the parameter of the given curve, which lead to the following simplifications: <math>\;|\vec c'(s)|=1\;</math> and  <math>\;\vec c''(s)=\kappa(s)\vec n(s)\;</math>, with <math>\kappa</math> the [[curvature]] and <math>\vec n</math> the unit normal. One gets for the involute:

:<math>\vec C_a(s)=\vec c(s) -\vec c'(s)(s-a)\ </math> and
:<math>\vec C_a'(s)=-\vec c''(s)(s-a)=-\kappa(s)\vec n(s)(s-a)\; </math>
and the statement: 
*At point <math> \vec C_a(a)</math>  the involute is ''not regular'' (because <math>| \vec C_a'(a)|=0</math> ),
and from  <math>\; \vec C_a'(s)\cdot\vec c'(s)=0 \;</math> follows:
* The normal of the involute at point <math>\vec C_a(s)</math> is the tangent of the given curve at point <math>\vec c(s)</math>.
* The involutes are [[parallel curve]]s, because of <math>\vec C_a(s)=\vec C_0(s)+a\vec c'(s)</math> and the fact, that <math>\vec c'(s)</math> is the unit normal at <math>\vec C_0(s)</math>.
The family of involutes and the family of tangents to the original curve makes up an [[Orthogonal coordinates|orthogonal coordinate system]]. Consequently, one may construct involutes graphically. First, draw the family of tangent lines. Then, an involute can be constructed by always staying orthogonal to the tangent line passing the point.

=== Cusps ===
This section is based on.<ref>{{Cite book |last=Arnolʹd |first=V. I. |url=https://www.worldcat.org/oclc/21873606 |title=Huygens and Barrow, Newton and Hooke : pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals |date=1990 |publisher=Birkhaüser Verlag |isbn=0-8176-2383-3 |location=Basel |oclc=21873606}}</ref>

There are generically two types of cusps in involutes. The first type is at the point where the involute touches the curve itself. This is a cusp of order 3/2. The second type is at the point where the curve has an inflection point. This is a cusp of order 5/2.

This can be visually seen by constructing a map <math>f: \R^2 \to \R^3</math> defined by <math display="block">(s, t) \mapsto (x(s) + t\cos(\theta), y(s) + t\sin(\theta), t)</math>where <math>(x(s), y(s))</math> is the arclength parametrization of the curve, and <math>\theta</math> is the slope-angle of the curve at the point <math>(x(s), y(s))</math>. This maps the 2D plane into a surface in 3D space. For example, this maps the circle into the [[hyperboloid of one sheet]].

By this map, the involutes are obtained in a three-step process: map <math>\R</math> to <math>\R^2</math>, then to the surface in <math>\R^3</math>, then project it down to <math>\R^2</math> by removing the z-axis: <math display="block">s \mapsto (s, l- s) \mapsto f(s, l- s) \mapsto (f(s, l- s)_x, f(s, l- s)_y)</math>where <math>l</math> is any real constant.

Since the mapping <math>s \mapsto f(s, l-s)</math> has nonzero derivative at all <math>s\in \R</math>, cusps of the involute can only occur where the derivative of <math>s \mapsto f(s, l-s)</math> is vertical (parallel to the z-axis), which can only occur where the surface in <math>\R^3</math> has a vertical tangent plane.

Generically, the surface has vertical tangent planes at only two cases: where the surface touches the curve, and where the curve has an inflection point.

==== cusp of order 3/2 ====

For the first type, one can start by the involute of a circle, with equation<math display="block">\begin{align}
X(t) &= r(\cos t + (t - a)\sin t)\\
Y(t) &= r(\sin t - (t - a)\cos t)
\end{align}</math>then set <math>a = 0</math>, and expand for small <math>t</math>, to obtain<math display="block">\begin{align}
X(t) &= r + r t^2/2 + O(t^4)\\
Y(t) &= rt^3/3 + O(t^5)
\end{align}</math>thus giving the order 3/2 curve <math>Y^2 - \frac{8}{9r} (X-r)^{3} + O(Y^{8/3}) = 0 </math>, a [[semicubical parabola]].

==== cusp of order 5/2 ====
[[File:Involutes of a cubic curve.svg|thumb|Tangents and involutes of the cubic curve <math>y = x^3</math>. The cusps of order 3/2 are on the cubic curve, while the cusps of order 5/2 are on the x-axis (the tangent line at the inflection point).]]
For the second type, consider the curve <math>y = x^3</math>. The arc from <math>x= 0</math>  to <math>x = s</math> is of length <math>\int_0^s \sqrt{1 + (3t^2)^2}dt = s + \frac{9}{10} s^5 - \frac 98 s^9 + O(s^{13})</math>, and the tangent at <math>x = s</math> has angle <math>\theta = \arctan(3s^2)</math>. Thus, the involute starting from <math>x= 0</math> at distance <math>L</math> has parametric formula<math display="block">\begin{cases}
    x(s) = s + (L-s-\frac{9}{10}s^5 + \cdots)\cos\theta \\
    y(s) = s^3 + (L-s-\frac{9}{10}s^5 + \cdots)\sin\theta
\end{cases}</math>Expand it up to order <math>s^5</math>, we obtain<math display="block">\begin{cases}
    x(s) = L - \frac 92 L s^4 + (\frac 92 L - \frac{9}{10}) s^5 + O(s^6)\\
    y(s) = 3Ls^2 - 2 s^3 + O(s^6)
\end{cases}</math>which is a cusp of order 5/2. Explicitly, one may solve for the polynomial expansion satisfied by <math>x, y</math>:<math display="block">\left(x - L + \frac{y^2}{2L} \right)^2 - \left(\frac 92 L + \frac{51}{10} \right)^2 \left(\frac{y}{3L} \right)^5 + O(s^{11}) = 0</math>or <math display="block">x = L - \frac{y^2}{2L} \pm \left(\frac 92 L + \frac{51}{10} \right) \left(\frac{y}{3L} \right)^{2.5} + O(y^{2.75}),\quad \quad y \geq 0 </math>which clearly shows the cusp shape.

Setting <math>L=0</math>, we obtain the involute passing the origin. It is special as it contains no cusp. By serial expansion, it has parametric equation<math display="block">\begin{cases}
    x(s) = \frac{18}{5} s^5 - \frac{126}{5} s^9 + O(s^{13}) \\
    y(s) = -2s^3 + \frac{54}{5} s^7 - \frac{318}{5} s^{11} + O(s^{15})
\end{cases}</math>or <math>x = -\frac{18}{5 \cdot 2^{1/3}}y^{5/3} + O(y^3)</math>