Ovoid (source code)
= Ovoid
{title2=$|\mathcal O|=q^2+1$}
In finite <projective space> $\operatorname{PG}(3,q)$ with $q>2$, an <ovoid> is a set of $q^2+1$ points with no three <collinear>. At each of its points there is a unique tangent plane meeting the set only at that point. It is a finite-geometric analogue of a strictly convex surface, although a definition using real convexity does not make sense over a <finite field>.<Some p-ranks Related to Orthogonal Spaces by Aart Blokhuis and G. Eric Moorhouse>[section 4, concluding discussion, manuscript page 23]{ref}
The <elliptic quadric> supplies examples. Over $\mathbb F_3$, the equation $x_0^2+x_1^2+x_2^2=x_3^2$ defines ten projective points. In the chart $x_3=1$, exactly one of the first three coordinates is nonzero, giving six points. At infinity all three are nonzero, giving four more after quotienting by simultaneous sign. Thus the count is $6+4=3^2+1$. This example links the incidence condition to a concrete <homogeneous polynomial> equation.
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