# Oval

An **oval** (from Latin *ovum*, "egg") is a closed curve in a plane which "loosely" resembles the outline of an egg. The term is not very specific, but in some areas (projective geometry, technical drawing, etc.) it is given a more precise definition, which may include either one or two axes of symmetry. In common English, the term is used in a broader sense: any shape which reminds one of an egg. The three-dimensional version of an oval is called an **ovoid**.

## Oval in geometry

The term **oval** when used to describe curves in geometry is not well-defined, except in the context of projective geometry. Many distinct curves are commonly called ovals or are said to have an "oval shape". Generally, to be called an oval, a plane curve should *resemble* the outline of an egg or an ellipse. In particular, these are common traits of ovals:

- they are differentiable (smooth-looking),[1] simple (not self-intersecting), convex, closed, plane curves;
- their shape does not depart much from that of an ellipse, and
- an oval would generally have an axis of symmetry, but this is not required.

Here are examples of ovals described elsewhere:

- Cassini ovals
- portions of some elliptic curves
- Moss's Egg
- superellipse
- Cartesian oval
- stadium

An **ovoid** is the surface in 3-dimensional space generated by rotating an oval curve about one of its axes of symmetry.
The adjectives **ovoidal** and **ovate** mean having the characteristic of being an ovoid, and are often used as synonyms for "egg-shaped".

## Projective geometry

- In a projective plane a set Ω of points is called an
**oval**, if:

- Any line l meets Ω in at most two points, and
- For any point
*P*∈ Ω there exists exactly one tangent line t through P, i.e.,*t*∩ Ω = {*P*}.

For *finite* planes (i.e. the set of points is finite) there is a more convenient characterization:[2]

- For a finite projective plane of
*order*n (i.e. any line contains*n*+ 1 points) a set Ω of points is an oval if and only if |Ω| =*n*+ 1 and no three points are collinear (on a common line).

An **ovoid** in a projective space is a set Ω of points such that:

- Any line intersects Ω in at most 2 points,
- The tangents at a point cover a hyperplane (and nothing more), and
- Ω contains no lines.

In the *finite* case only for dimension 3 there exist ovoids. A convenient characterization is:

- In a 3-dim. finite projective space of order
*n*> 2 any pointset Ω is an ovoid if and only if |Ω| and no three points are collinear.[3]

## Egg shape

The shape of an egg is approximated by the "long" half of a prolate spheroid, joined to a "short" half of a roughly spherical ellipsoid, or even a slightly oblate spheroid. These are joined at the equator and share a principal axis of rotational symmetry, as illustrated above. Although the term *egg-shaped* usually implies a lack of reflection symmetry across the equatorial plane, it may also refer to true prolate ellipsoids. It can also be used to describe the 2-dimensional figure that, if revolved around its major axis, produces the 3-dimensional surface.

## Technical drawing

In technical drawing, an **oval** is a figure constructed from two pairs of arcs, with two different radii (see image on the right). The arcs are joined at a point in which lines tangential to both joining arcs lie on the same line, thus making the joint smooth. Any point of an oval belongs to an arc with a constant radius (shorter or longer), but in an ellipse, the radius is continuously changing.

## In common speech

In common speech, "oval" means a shape rather like an egg or an ellipse, which may be two-dimensional or three-dimensional. It also often refers to a figure that resembles two semicircles joined by a rectangle, like a cricket infield, speed skating rink or an athletics track. However, this is more correctly called a stadium. Sometimes, it can even refer to any rectangle with rounded corners.

The terms "ellipse" and "oblong" are often used interchangibly with oval, despite not being precise synonyms.[4][5]

## See also

- Ellipse
- Stadium (geometry)
- Vesica piscis – a pointed oval

## Notes

- If the property makes sense: on a differentiable manifold. In more general settings one might require only a unique tangent line at each point of the curve.
- Dembowski 1968, p. 147
- Dembowski 1968, p. 48
- "Definition of ellispe in US English by Oxford Dictionaries".
*New Oxford American Dictionary*. Oxford University Press. Retrieved 9 July 2018. - "Definition of oblong in US English by Oxford Dictionaries".
*New Oxford American Dictionary*. Oxford University Press. Retrieved 9 July 2018.

- Dembowski, Peter (1968),
*Finite geometries*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44, Berlin, New York: Springer-Verlag, ISBN 3-540-61786-8, MR 0233275