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A curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036720/e0367201.png" /> assigned to the plane curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036720/e0367202.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036720/e0367203.png" /> is the [[Evolute|evolute]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036720/e0367204.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036720/e0367205.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036720/e0367206.png" /> is the arc length parameter of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036720/e0367207.png" />) is the equation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036720/e0367208.png" />, then the equation of its evolvent has the form
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''involute''
  
<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/e036720/e0367209.png" /></td> </tr></table>
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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
 
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$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036720/e03672010.png" /> is an arbitrary constant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036720/e03672011.png" /> the unit tangent vector to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036720/e03672012.png" />. The figures show the construction of the evolvent in two typical cases: a) if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036720/e03672013.png" /> the curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036720/e03672014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036720/e03672015.png" /> does not vanish (the evolvent is a regular curve); and b) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036720/e03672016.png" /> vanishes only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036720/e03672017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036720/e03672018.png" /> (the point corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036720/e03672019.png" /> on the evolvent is a cusp of the second kind).
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\bar{\mathbf{r}} = \mathbf{r}(s) + (c-s)\tau(s) \,,
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$$
 +
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" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/e036720a.gif" />
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Figure: e036720b
 
Figure: e036720b
  
About the evolvent of a surface, see [[Evolute (surface)|Evolute (surface)]].
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About the evolvent of a surface, see [[Evolute (surface)]].
  
  
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The evolvent is often called the involute of the curve. Involvents play a part in the construction of gears.
 
The evolvent is often called the involute of the curve. Involvents play a part in the construction of gears.
  
For references see also [[Evolute|Evolute]].
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For references see also [[Evolute]].
  
 
====References====
 
====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>
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<table>
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<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>
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</table>
 +
 
 +
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Revision as of 21:08, 17 December 2017

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).

Figure: e036720a

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

[a1] K. Strubecker, "Differential geometry" , I , de Gruyter (1964)
[a2] M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) pp. 305ff (Translated from French)
[a3] J.L. Coolidge, "A treatise on algebraic plane curves" , Dover, reprint (1959) pp. 195
[a4] H.W. Guggenheimer, "Differential geometry" , McGraw-Hill (1963) pp. 25; 60
[a5] M. Berger, "Geometry" , I , Springer (1987) pp. 253–254
How to Cite This Entry:
Evolvent of a plane curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Evolvent_of_a_plane_curve&oldid=42552
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article