In projective geometry an ovoid is a sphere like pointset (surface) in a projective space of dimension d ≥ 3. Simple examples in a real projective space are hyperspheres (quadrics). The essential geometric properties of an ovoid O {\displaystyle {\mathcal {O}}} are:
Any line intersects O {\displaystyle {\mathcal {O}}} in at most 2 points, The tangents at a point cover a hyperplane (and nothing more), and
O {\displaystyle {\mathcal {O}}} contains no lines. Property 2) excludes degenerated cases (cones,...). Property 3) excludes ruled surfaces (hyperboloids of one sheet, ...). An ovoid is the spatial analog of an oval in a projective plane. An ovoid is a special type of a quadratic set. Ovoids play an essential role in constructing examples of Möbius planes and higher dimensional Möbius geometries.
Definition of an ovoid In a projective space of dimension d ≥ 3 a set O {\displaystyle {\mathcal {O}}} of points is called an ovoid, if (1) Any line g meets O {\displaystyle {\mathcal {O}}} in at most 2 points. In the case of | g ∩ O | = 0 {\displaystyle |g\cap {\mathcal {O}}|=0} , the line is called a passing (or exterior) line, if | g ∩ O | = 1 {\displaystyle |g\cap {\mathcal {O}}|=1} the line is a tangent line, and if | g ∩ O | = 2 {\displaystyle |g\cap {\mathcal {O}}|=2} the line is a secant line.
(2) At any point P ∈ O {\displaystyle P\in {\mathcal {O}}} the tangent lines through P cover a hyperplane, the tangent hyperplane, (i.e., a projective subspace of dimension d − 1). (3) O {\displaystyle {\mathcal {O}}} contains no lines. From the viewpoint of the hyperplane sections, an ovoid is a rather homogeneous object, because
For an ovoid O {\displaystyle {\mathcal {O}}} and a hyperplane ε {\displaystyle \varepsilon } , which contains at least two points of O {\displaystyle {\mathcal {O}}} , the subset ε ∩ O {\displaystyle \varepsilon \cap {\mathcal {O}}} is an ovoid (or an oval, if d = 3) within the hyperplane ε {\displaystyle \varepsilon } . For finite projective spaces of dimension d ≥ 3 (i.e., the point set is finite, the space is pappian), the following result is true:
If O {\displaystyle {\mathcal {O}}} is an ovoid in a finite projective space of dimension d ≥ 3, then d = 3. (In the finite case, ovoids exist only in 3-dimensional spaces.) In a finite projective space of order n >2 (i.e. any line contains exactly n + 1 points) and dimension d = 3 any pointset O {\displaystyle {\mathcal {O}}} is an ovoid if and only if | O | = n 2 + 1 {\displaystyle |{\mathcal {O}}|=n^{2}+1} and no three points are collinear (on a common line). Replacing the word projective in the definition of an ovoid by affine, gives the definition of an affine ovoid. If for an (projective) ovoid there is a suitable hyperplane ε {\displaystyle \varepsilon } not intersecting it, one can call this hyperplane the hyperplane ε ∞ {\displaystyle \varepsilon _{\infty }} at infinity and the ovoid becomes an affine ovoid in the affine space corresponding to ε ∞ {\displaystyle \varepsilon _{\infty }} . Also, any affine ovoid can be considered a projective ovoid in the projective closure (adding a hyperplane at infinity) of the affine space.
Examples
In real projective space (inhomogeneous representation)
O = { ( x 1 , . . . , x d ) ∈ R d | x 1 2 + ⋯ + x d 2 = 1 } , {\displaystyle {\mathcal {O}}=\{(x_{1},...,x_{d})\in {\mathbb {R} }^{d}\;|\;x_{1}^{2}+\cdots +x_{d}^{2}=1\}\ ,} (hypersphere)
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