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Over a finite field ${\mathbb{F}}_q$, an elliptic quadric $Q^{-}(2n+1,q)$ is the zero set in projective space $\mathrm{PG}(2n+1,q)$ of a nonsingular quadratic form on a $(2n+2)$-dimensional vector space whose Witt index is $n$. Thus the largest vector subspaces on which the form vanishes identically have dimension $n$, one less than for a hyperbolic form in the same dimension. The word elliptic describes this type of quadratic form.

The especially important case $Q^{-}(3,q)$ contains $q^2+1$ points and no projective line. A line therefore meets it in at most two points, making it an example of an ovoid. These properties make elliptic quadrics useful configurations in finite geometry and in the study of incidence ranks.^[ref]

For example, in $\mathrm{PG}(3,3)$ the equation $x_0^2+x_1^2+x_2^2=x_3^2$ gives an elliptic quadric. With $x_3=1$ there are six solutions, since exactly one of the first three coordinates is nonzero. With $x_3=0$ all three must be nonzero; the eight vectors give four projective points after identifying scalar multiples. There are ten points altogether, as $3^2+1$ predicts.

In finite projective space $\mathrm{PG}(3,q)$ 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.^[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.