  We study solutions of a homogeneous quadratic equation $q(x_0,\dots, x_n)=0$,
defined over a field $K$, where the $x_i$ are themselves homogeneous
polynomials of some degree $d$ in $r+1$ variables. Equivalently, we are looking
at rational maps from projective $r$-space $P^r$ to a quadric hypersurface $Q$,
defined over a field $K$.
  The space of maps of $P^1$ to a quadric $Q$ is stably birational to $Q$ if
$d$ is even and to the orthogonal Grassmannian of lines in $Q$ if $d$ is odd.
Most of the paper is devoted to obtaining similar descriptions for the spaces
parametrizing maps of $P^2$ to quadrics, given by degree 2 polynomials. The
most interesting case is 4-dimensional quadrics when there are 5 irreducible
components. The methods are mostly classical, involving the Veronese surface,
its equations and projections.
  In the real case, these results provide some of the last steps of a project,
started by Kummer and Darboux, to describe all surfaces that contain at least 2
circles through every point.
