Oval

Projective geometry

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

1. Any line l meets Ω in at most two points, and
2. 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:

• 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:

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

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

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

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 two-dimensional figure that, if revolved around its major axis, produces the three-dimensional surface.

Mathematical definition of an oval

Based on the meaning of the word "oval" (i.e., "egg"), an oval refers to a flat representation of an egg. Since there are a variety of egg shapes in nature, a mathematical description should graphically reproduce the oval shape of any egg profile. Four key dimensions (measurements) can be adopted as the defining parameters of an oval:

• length L,
• maximum breadth B,
• the shift w of the maximum breadth from the center, that is, from the point x = L/2, and
• the breadth D_p at a point ⁠1/4⁠ of the length from the pointed end.

Accordingly, the mathematical formula for an oval can be written as follows: $${\displaystyle y=\pm \frac{B}{2}{\sin }^n\left(\frac{\pi (1-p)x}{x-pL}\right),}$$ in which $${\displaystyle p=\frac{1}{2}\left[1+\frac{1}{2}{\left(\frac{w}{L}\right)}^{-1}\right],}$$ $${\displaystyle n=\frac{\ln \left(\frac{D_p}{B}\right)}{\ln \left[\sin \left(\pi \frac{1-p}{1-4p}\right)\right]}.}$$

It is more convenient to operate with various modifications of oval shapes when a unit length (L = 1) is used, and the oval parameters are presented as their combinations conventionally expressed by indices, that is, the shape index (B/L), the asymmetry index (w/L), and the conicity index (D_p/B). Herewith, the mathematical formula for a unit oval can be written as: $${\displaystyle \frac{y}{L}=\pm \frac{1}{2}\cdot \frac{B}{L}{\sin }^n\left(\pi \frac{1-p}{\frac{x}{L}-p}\cdot \frac{x}{L}\right).}$$

The variety of index values allows for countless variations of oval shapes, from standard geometric shapes such as the circle and ellipse, to classic egg-shaped ovals and even more exotic pyriform (pear-shaped) and biconical shapes.

Certain values of the conicity index allow to transform a classical ellipse into a superellipse:

Technical drawing

In technical drawing, an oval is a figure that is 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 most correctly called a stadium.

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