Editing Involute (section)

==== 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>