View source for Evolvent of a plane curve

{{TEX|done}}{{MSC|53A04}}

''involute''

A curve $\bar\gamma$ assigned to the plane curve $\gamma$ such that $\gamma$ is the [[evolute]] of $\bar\gamma$. If $\mathbf{r} = \mathbf{r}(s)$ (where $s$ is the arc length parameter of $\gamma$) is the equation of $\gamma$, then the equation of its evolvent has the form
$$
\bar{\mathbf{r}} = \mathbf{r}(s) + (c-s)\tau(s) \,,
$$
where $c$ is an arbitrary constant and $\tau$ the unit tangent vector to $\gamma$. The figures show the construction of the evolvent in two typical cases: a) if for any $s<c$ the [[curvature]] $k(s)$ of $\gamma$ does not vanish (the evolvent is a regular curve); and b) if $k(s)$ vanishes only for $s=s_1$ and $k'(s_1) \ne 0$ (the point corresponding to $s=s_1$ on the evolvent is a cusp of the second kind).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/e036720a.gif" />

Figure: e036720a

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/e036720b.gif" />

Figure: e036720b

About the evolvent of a surface, see [[Evolute (surface)]].



====Comments====
The evolvent is often called the involute of the curve. Involvents play a part in the construction of gears.

For references see also [[Evolute]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Strubecker,   "Differential geometry" , '''I''' , de Gruyter  (1964)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Berger,   B. Gostiaux,   "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  pp. 305ff  (Translated from French)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  J.L. Coolidge,   "A treatise on algebraic plane curves" , Dover, reprint  (1959)  pp. 195</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  H.W. Guggenheimer,   "Differential geometry" , McGraw-Hill  (1963)  pp. 25; 60</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Berger,   "Geometry" , '''I''' , Springer  (1987)  pp. 253–254</TD></TR>
</table>