Hyperbolic plane (quadratic form, $q(xe+yf)=xy$)

A hyperbolic plane is a two-dimensional vector space with a quadratic form and a basis $e,f$ satisfying $q(e)=q(f)=0$ and $B(e,f)=1$, where $B$ is the polar bilinear form of a quadratic form. Equivalently, $q(xe+yf)=xy$. This definition also applies in characteristic two. Over the real numbers, the identity $xy=((x+y)/2{)}^2-((x-y)/2{)}^2$ gives one positive and one negative square, hence signature zero. The signature statement is specific to real quadratic forms.