Ovoid (polar space)
In mathematics, an ovoid "O" of a (finite) polar space of rank "r" is a set of points, such that every subspace of rank formula_1 intersects "O" in exactly one point. Cases. Symplectic polar space. An ovoid of formula_2 (a symplectic polar space of rank "n") would contain formula_3 points. However it only has an ovoid if and only formula_4 and "q" is even. In that case, when the polar space is embedded into formula_5 the classical way, it is also an ovoid in the projective geometry sense. Hermitian polar space. Ovoids of formula_6 and formula_7 would  contain formula_8 points. Hyperbolic quadrics. An ovoid of a hyperbolic quadricformula_9would contain formula_10 points. Parabolic quadrics. An ovoid of a parabolic quadric formula_11 would contain formula_3 points. For formula_4, it is easy to see to obtain an ovoid by cutting the parabolic quadric with a hyperplane, such that the intersection is an elliptic quadric. The intersection is an ovoid. If "q" is even, formula_14 is isomorphic (as polar space) with formula_2, and thus due to the above, it has no ovoid for formula_16. Elliptic quadrics. An ovoid of an elliptic quadric formula_17would contain formula_18 points.