View source for Oval

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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 $E$ 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|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(2)|ovoid]].

====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>