Reduction algorithm for a positive definite binary quadratic form

For a positive definite integral form $[a,b,c]$, a unimodular shear can replace $b$ by its representative $b^{\prime }$ with $|b^{\prime }|\le a$. If the resulting $c^{\prime }=(b^{\prime 2}-D)/(4a)$ is smaller than $a$, swapping the variables replaces the leading coefficient by the smaller positive integer $c^{\prime }$. Repeating terminates and produces $|b|\le a\le c$; determinant-one sign changes enforce the boundary convention. Thus every positive definite integral binary quadratic form is properly equivalent to a reduced form.