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Modulus (Quadratic Invariants)

The quantity $ps-rq$ obtained by letting

$\displaystyle x$ $\textstyle =$ $\displaystyle pX+qY$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle rX+sY$ (2)

in
\begin{displaymath} ax^2+2bxy+cy^2 \end{displaymath} (3)

so that
$\displaystyle A$ $\textstyle =$ $\displaystyle ap^2+2bpr+cr^2$ (4)
$\displaystyle B$ $\textstyle =$ $\displaystyle apq+b(ps+qr)+crs$ (5)
$\displaystyle C$ $\textstyle =$ $\displaystyle aq^2+2bqs+cs^2$ (6)

and
\begin{displaymath} B^2-AC=(ps-rq)^2(b^2-ac), \end{displaymath} (7)

is called the modulus.



© 1996-9 Eric W. Weisstein
1999-05-26