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A closed convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070640/o0706401.png" />-smooth curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070640/o0706402.png" />. The points of an oval at which the [[Curvature|curvature]] is extremal are called the vertices of the oval. The number of vertices is at least four.
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A closed convex $C^2$-smooth curve in $\R^2$. The points of an oval at which the [[Curvature|curvature]] is extremal are called the vertices of the oval. The number of vertices is at least four.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070640/o0706403.png" /> be an oval, traversed counter-clockwise, in the plane with rectangular Cartesian coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070640/o0706404.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070640/o0706405.png" /> be the distance from the origin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070640/o0706406.png" /> to the directed tangent line to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070640/o0706407.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070640/o0706408.png" /> if the rotation of the tangent line relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070640/o0706409.png" /> is counter-clockwise). Then the equation of the tangent line is
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Let $E$ be an oval, traversed counter-clockwise, in the plane with rectangular Cartesian coordinates $x,y$ let $h$ be the distance from the origin $O$ to the directed tangent line to $E$ ($h>0$ if the rotation of the tangent line relative to $O$ is counter-clockwise). Then the equation of the tangent line is
  
<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/o/o070/o070640/o07064010.png" /></td> </tr></table>
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$$x\cos\tau + y\sin\tau=h(\tau),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070640/o07064011.png" /> is the angle made by the tangent line and the axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070640/o07064012.png" />. The quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070640/o07064013.png" /> is called the support function of the oval. The radius of curvature of the oval is
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where $\tau$ is the angle made by the tangent line and the axis $Ox$. The quantity $h(\tau)$ is called the support function of the oval. The radius of curvature of the oval is
  
<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/o/o070/o070640/o07064014.png" /></td> </tr></table>
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$$r=h+\frac{d^2 h}{d\tau^2};$$
  
 
and the length of the oval (Cauchy's formula) is
 
and the length of the oval (Cauchy's formula) is
  
<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/o/o070/o070640/o07064015.png" /></td> </tr></table>
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$$L=\int\limits_{-\pi}^\pi h(\tau)d\tau.$$
  
The following isoperimetric inequality holds for the length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070640/o07064016.png" /> and the area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070640/o07064017.png" /> of the region inside the oval:
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The following isoperimetric inequality holds for the length $L$ and the area $F$ of the region inside the oval:
  
<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/o/o070/o070640/o07064018.png" /></td> </tr></table>
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$$L^2-4\pi F\geq 0$$
  
 
(for more details see [[Bonnesen inequality|Bonnesen inequality]]).
 
(for more details see [[Bonnesen inequality|Bonnesen inequality]]).
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====Comments====
 
====Comments====
Sometimes smoothness is not assumed, so that any closed convex curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070640/o07064019.png" /> is called an oval. In finite (projective) geometry the term  "oval"  denotes a special kind of [[Ovoid(2)|ovoid]].
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Sometimes smoothness is not assumed, so that any closed convex curve in $\R^2$ is called an oval. In finite (projective) geometry the term  "oval"  denotes a special kind of [[Ovoid(2)|ovoid]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.P. Do Carmo,  "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S.S. Chern,  "Curves and surfaces in Euclidean space" , Prentice-Hall  (1967)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  T. Bonnesen,  W. Fenchel,  "Theorie der konvexen Körper" , Springer  (1934)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.P. Do Carmo,  "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S.S. Chern,  "Curves and surfaces in Euclidean space" , Prentice-Hall  (1967)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  T. Bonnesen,  W. Fenchel,  "Theorie der konvexen Körper" , Springer  (1934)</TD></TR></table>

Revision as of 10:02, 12 March 2012

A closed convex $C^2$-smooth curve in $\R^2$. The points of an oval at which the curvature is extremal are called the vertices of the oval. The number of vertices is at least four.

Let $E$ be an oval, traversed counter-clockwise, in the plane with rectangular Cartesian coordinates $x,y$ let $h$ be the distance from the origin $O$ to the directed tangent line to $E$ ($h>0$ if the rotation of the tangent line relative to $O$ is counter-clockwise). Then the equation of the tangent line is

$$x\cos\tau + y\sin\tau=h(\tau),$$

where $\tau$ is the angle made by the tangent line and the axis $Ox$. The quantity $h(\tau)$ is called the support function of the oval. The radius of curvature of the oval is

$$r=h+\frac{d^2 h}{d\tau^2};$$

and the length of the oval (Cauchy's formula) is

$$L=\int\limits_{-\pi}^\pi h(\tau)d\tau.$$

The following isoperimetric inequality holds for the length $L$ and the area $F$ of the region inside the oval:

$$L^2-4\pi F\geq 0$$

(for more details see Bonnesen inequality).


Comments

Sometimes smoothness is not assumed, so that any closed convex curve in $\R^2$ is called an oval. In finite (projective) geometry the term "oval" denotes a special kind of ovoid.

References

[a1] M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a2] M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976)
[a3] S.S. Chern, "Curves and surfaces in Euclidean space" , Prentice-Hall (1967)
[a4] T. Bonnesen, W. Fenchel, "Theorie der konvexen Körper" , Springer (1934)
How to Cite This Entry:
Oval. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Oval&oldid=15439
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article