= Hyperbolic plane
{disambiguate=quadratic form}
{title2=$q(xe+yf)=xy$}
= Hyperbolic plane of a quadratic form
{synonym}
= Hyperbolic planes of a quadratic form
{synonym}
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.
Back to article page
