Editing Involute (section)

== Involute of a parameterized curve ==
{{See also|Arc length}}
Let   <math>  \vec c(t),\; t\in [t_1,t_2] </math> be a [[regular curve]] in the plane with its [[Curvature (mathematics)|curvature]] nowhere 0 and <math>a\in (t_1,t_2)</math>, then the curve with the parametric representation

<math>\vec C_a(t)=\vec c(t) -\frac{\vec c'(t)}{|\vec c'(t)|}\; \int_a^t|\vec c'(w)|\; dw </math>

is an ''involute'' of the given curve.
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|{{Show|Proof|The string acts as a [[tangent (geometry)|tangent]] to the curve <math> \vec c(t) </math>. Its length is changed by an amount equal to the [[arc length]] traversed as it winds or unwinds. Arc length of the curve traversed in the interval <math>[a,t]</math> is given by

<math>  \int_a^t|\vec c'(w)|\; dw </math> 

where <math> a </math> is the starting point from where the arc length is measured. Since the tangent vector depicts the taut string here, we get the string vector as

<math>\frac{\vec c'(t)}{|\vec c'(t)|}\; \int_a^t|\vec c'(w)|\; dw </math>

The vector corresponding to the end point of the string (<math>\vec C_a(t) </math>) can be easily calculated using [[Vector notation#Operations|vector addition]], and one gets

<math>\vec C_a(t)=\vec c(t) -\frac{\vec c'(t)}{|\vec c'(t)|}\; \int_a^t|\vec c'(w)|\; dw </math>
}}
|}

Adding an arbitrary but fixed number <math>l_0</math> to the integral <math> \Bigl(\int_a^t|\vec c'(w)|\; dw\Bigr) </math> results in an involute corresponding to a string extended by <math>l_0</math> (like a ball of wool [[yarn]] having some length of thread already hanging before it is unwound). Hence, the involute can be varied by constant <math>a</math> and/or adding a number to the integral (see [[#Involutes of a semicubic parabola|Involutes of a semicubic parabola]]).

If <math>\vec c(t)=(x(t),y(t))^T</math> one gets

:<math>\begin{align}
X(t) &= x(t) - \frac{x'(t)}{\sqrt{x'(t)^2 + y'(t)^2}} \int_a^t \sqrt{x'(w)^2 + y'(w)^2} \,dw \\
Y(t) &= y(t) - \frac{y'(t)}{\sqrt{x'(t)^2 + y'(t)^2}} \int_a^t \sqrt{x'(w)^2 + y'(w)^2} \,dw \; .
\end{align}</math>