thumb|This oval, with only one [[axis of symmetry, resembles a chicken egg.]]
An oval () is a closed curve in a plane which resembles the outline of an egg. 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 an ellipse. 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. simple (not self-intersecting), convex, closed, and plane curves;
- their shape does not depart much from that of an ellipse; and
thumb|To the definition of an ovoid.
- an oval typically has 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 three-dimensional space generated by rotating an oval curve about an axis of symmetry.
The adjectives ovoidal and ovate mean having the characteristic of being an ovoid, and are often used as synonyms for "egg-shaped".
thumb|To the definition of an oval in a [[projective plane.]]
Projective geometry
- In a projective plane a set of points is called an oval, if:
- Any line meets in at most two points, and
- For any point , there exists exactly one tangent line through , i.e., }.
For finite planes (i.e., the set of points is finite) there is a more convenient characterization:
- For a finite projective plane of order (i.e., any line contains points) a set of points is an oval if and only if and no three points are collinear (on a common line).
An ovoid in a projective space is a set of points such that:
- Any line intersects in at most 2 points,
- The tangents at a point cover a hyperplane (and nothing more), and
- contains no lines.
thumb|An oval [[egg shape.]]
In the finite case only for dimension 3, there exist ovoids. A convenient characterization is:
- In a 3-dimensional finite projective space of order any pointset is an ovoid if and only if ||<math>=n^2+1</math> 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.
thumb|Oval and its four key [[Geometry|geometrical parameters: 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<sub>p</sub> at a point of the length from the pointed end.]]
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<sub>p</sub> at a point of the length from the pointed end.
Accordingly, the mathematical formula for an oval can be written as follows: 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.
See also
- Ellipse
- Ellipsoidal dome
- Stadium (geometry)
- Symbolism of domes
- Vesica piscis – a pointed oval