Ovoid

Remarks

The hyperplane in clause 2 is the tangent hyperplane of $O$ at $P$. In a projective plane the same two conditions define an oval.

Example: in $\mathrm{PG}(3, q)$ over the finite field $\F_q$, an elliptic quadric is an ovoid. It contains no line, so a line meets it in at most two points. At each point, the tangent lines are the lines through that point in its tangent plane. It has $q^2 + 1$ points. In real projective $3$-space, a sphere is an ovoid.

Non-example: a hyperbolic quadric $x_0 x_1 = x_2 x_3$ contains whole lines, such as $x_0 = x_2 = 0$, so clause 1 fails.

In $\mathrm{PG}(3, q)$ every ovoid has $q^2 + 1$ points. For $q = 2^{2e+1}$ with $e \ge 1$, Tits found ovoids that are not quadrics. In finite projective spaces of dimension at least $4$ there are no ovoids.