Oval
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An oval (from
Oval in geometry
The term oval when used to describe
- 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
Technical drawing
In
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
The term "ellipse" is often used interchangeably with oval, despite not being a precise synonym.[4] The term "oblong" is often used incorrectly to describe an elongated oval or 'stadium' shape.[5] However, in geometry, an oblong is a rectangle with unequal adjacent sides (i.e., not a square).[6]
See also
- Ellipse
- Ellipsoidal dome
- Stadium (geometry)
- Vesica piscis – a pointed oval
- Symbolism of domes
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 ellipse in US English by Oxford Dictionaries". New Oxford American Dictionary. Oxford University Press. Archived from the original on September 27, 2016. Retrieved 9 July 2018.
- ^ "Definition of oblong in US English by Oxford Dictionaries". New Oxford American Dictionary. Oxford University Press. Archived from the original on September 24, 2016. Retrieved 9 July 2018.
- ^ "Definition of quadliraterals, Clark University, Dept. of Maths and Computer Science". Clark University, Definitions of quadrilaterals. Retrieved 21 October 2020.
- Dembowski, Peter (1968), Finite geometries, MR 0233275