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''of a plane curve''
 
''of a plane curve''
  
The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e0366701.png" /> of the centres of curvature of the given curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e0366702.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e0366703.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e0366704.png" /> is the arc length parameter of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e0366705.png" />) is the equation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e0366706.png" />, then the equation of its evolute has the form
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e0366707.png" /></td> </tr></table>
+
$$
 +
\widetilde{\mathbf r}  = \mathbf r +
 +
\frac{1}{k}
 +
{\pmb\nu } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e0366708.png" /> is the curvature and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e0366709.png" /> the unit normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e03667010.png" />. The figures shows the construction of the evolute in three typical cases:
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e03667011.png" /> has a fixed sign and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e03667012.png" /> does not vanish;
+
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" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/e036670a.gif" />
Line 17: Line 44:
 
Figure: e036670b
 
Figure: e036670b
  
b) if along the entire curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e03667013.png" /> has a fixed sign and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e03667014.png" /> vanishes for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e03667015.png" />; and
+
b) if along the entire curve $  k  ^  \prime  $
 +
has a fixed sign and $  k $
 +
vanishes for $  s = s _ {0} $;  
 +
and
  
c) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e03667016.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e03667017.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e03667018.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e03667019.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e03667020.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e03667021.png" /> does not vanish (the point of the evolute corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e03667022.png" /> is a cusp).
+
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" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/e036670c.gif" />
Line 25: Line 62:
 
Figure: e036670c
 
Figure: e036670c
  
The length of the arc of the evolute corresponding to the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e03667023.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e03667024.png" /> is
+
The length of the arc of the evolute corresponding to the segment $  s _ {1} \leq  s \leq  s _ {2} $
 
+
of $  \gamma $
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e03667025.png" /></td> </tr></table>
+
is
  
The evolute is the envelope of the normals to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e03667026.png" />. The curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e03667027.png" /> is called the evolvent of its evolute (cf. [[Evolvent of a plane curve|Evolvent of a plane curve]]).
+
$$
 +
\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====
 
====Comments====
The evolvent is also called the involute; thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e03667028.png" /> is the evolute of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e03667029.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e03667030.png" /> is the involute of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036670/e03667031.png" />, cf. [[Evolvent of a plane curve|Evolvent of a plane curve]].
+
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====
 
====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>
 
<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>

Latest revision as of 19:38, 5 June 2020


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;

Figure: e036670a

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).

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).

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.

References

[a1] H.-R. Müller, "Kinematik" , de Gruyter (1963)
[a2] M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) pp. 305 (Translated from French)
[a3] J.L. Coolidge, "Algebraic plane curves" , Dover, reprint (1959) pp. 195
[a4] M. Berger, "Geometry" , I , Springer (1987) pp. 253–254
[a5] H.W. Guggenheimer, "Differential geometry" , McGraw-Hill (1963) pp. 25; 60
How to Cite This Entry:
Evolute. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Evolute&oldid=46863
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article