Anoval (fromLatin ovum 'egg') is aclosed curve in aplane which resembles the outline of anegg. The term is not very specific, but in some areas of mathematics(projective geometry,technical drawing, etc.) it is given a more precise definition, which may include either one or two axes of symmetry of anellipse. 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 anovoid.

The termoval when used to describecurves ingeometry is not well-defined, except in the context ofprojective geometry. Many distinct curves are commonly called ovals or are said to have an "oval shape". Generally, to be called an oval, aplane curve shouldresemble the outline of anegg or anellipse. In particular, these are common traits of ovals:

• they aredifferentiable (smooth-looking),^[1]simple (not self-intersecting),convex,closed,plane curves;
• theirshape does not depart much from that of anellipse, and
• an oval would generally have anaxis of symmetry, but this is not required.

Here are examples of ovals described elsewhere:

• Cassini ovals
• portions of someelliptic curves
• Moss's egg
• superellipse
• Cartesian oval
• stadium

Anovoid is the surface in 3-dimensional space generated by rotating an oval curve about one of its axes of symmetry.The adjectivesovoidal andovate mean having the characteristic of being an ovoid, and are often used assynonyms for "egg-shaped".

• In aprojective plane a setΩ of points is called anoval, if:

1. Any linel meetsΩ in at most two points, and
2. For any pointP ∈ Ω there exists exactly one tangent linet throughP, i.e.,t ∩ Ω = {P}.

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

• For a finite projective plane ofordern (i.e. any line containsn + 1 points) a setΩ of points is an oval if and only if|Ω| =n + 1 and no three points arecollinear (on a common line).

Anovoid in a projective space is a setΩ of points such that:

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

In thefinite case only for dimension 3 there exist ovoids. A convenient characterization is:

• In a 3-dim. finite projective space of ordern > 2 any pointsetΩ is an ovoid if and only if |Ω|${\displaystyle =n^2+1}$  and no three points are collinear.^[3]

The shape of anegg is approximated by the "long" half of a prolatespheroid, joined to a "short" half of a roughly sphericalellipsoid, or even a slightlyoblate spheroid. These are joined at the equator and share aprincipal axis ofrotational symmetry, as illustrated above. Although the termegg-shaped usually implies a lack ofreflection 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 itsmajor axis, produces the 3-dimensional surface.

Intechnical drawing, anoval is a figure that is constructed from two pairs of arcs, with two differentradii (see image on the right). The arcs are joined at a point in which linestangential 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 anellipse, the radius is continuously changing.

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 acricket infield,speed skating rink or anathletics track. However, this is most correctly called astadium.

The term "ellipse" is often used interchangeably with oval, but it has a more specific mathematical meaning.^[4] The term "oblong" is also used to mean oval,^[5] though in geometry an oblong refers to rectangle with unequal adjacent sides, not a curved figure.^[6]

• Ellipse
• Ellipsoidal dome
• Stadium (geometry)
• Vesica piscis – a pointed oval
• Symbolism of domes

• 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