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''of a plane curve''

The set  $  \widetilde \gamma   $
of the centres of curvature of the given curve  $  \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 evolute has the form

$$ 
\widetilde{\mathbf r}   =  \mathbf r + 
\frac{1}{k}
 {\pmb\nu } ,
$$

where  $  k $
is the curvature and  $  {\pmb\nu } $
the unit normal to  $  \gamma $. 
The figures shows the construction of the evolute in three typical cases:

a) if along the entire curve  $  k  ^  \prime  $
has a fixed sign and  $  k $
does not vanish;

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/e036670a.gif" />

Figure: e036670a

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/e036670b.gif" />

Figure: e036670b

b) if along the entire curve  $  k  ^  \prime  $
has a fixed sign and  $  k $
vanishes for  $  s = s _ {0} $; 
and

c) if  $  k  ^  \prime  > 0 $
for  $  s < s _ {0} $; 
$  k  ^  \prime  ( s) < 0 $
for  $  s > s _ {0} $; 
$  k  ^  \prime  ( s _ {0} ) = 0 $, 
and  $  k $
does not vanish (the point of the evolute corresponding to  $  s = s _ {0} $
is a cusp).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/e036670c.gif" />

Figure: e036670c

The length of the arc of the evolute corresponding to the segment  $  s _ {1} \leq  s \leq  s _ {2} $
of  $  \gamma $
is

$$ 
\widetilde{s}  ( s _ {1} , s _ {2} )  =  \left | 
\frac{1}{k ( s _ {2} ) }
 -

\frac{1}{k ( s _ {1} ) }
 \right | .
$$

The evolute is the envelope of the normals to  $  \gamma $. 
The curve  $  \gamma $
is called the evolvent of its evolute (cf. [[Evolvent of a plane curve|Evolvent of a plane curve]]).

====Comments====
The evolvent is also called the involute; thus, if  $  \gamma  ^  \prime  $
is the evolute of  $  \gamma $, 
then  $  \gamma $
is the involute of  $  \gamma  ^  \prime  $, 
cf. [[Evolvent of a plane curve|Evolvent of a plane curve]].

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