Ovale

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 (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. 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), 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". 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: 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 = 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.