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
and the length of the oval (Cauchy's formula) is
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).
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.
|[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)|
Oval. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Oval&oldid=31013