Cruciform

$\begin{figure}\begin{center}\BoxedEPSF{cruciform.epsf}\end{center}\end{figure}$

A plane curve also called the Cross Curve and Policeman on Point Duty Curve (Cundy and Rollett 1989). It is given by the equation

$\begin{displaymath} x^2y^2-a^2x^2-b^2y^2=0, \end{displaymath}$

(1)

which is equivalent to

$\begin{displaymath} 1-{a^2\over x^2}-{b^2\over y^2}=0 \end{displaymath}$

(2)

$\begin{displaymath} {a^2\over x^2}+{b^2\over y^2}=1, \end{displaymath}$

(3)

or, rewriting,

$\begin{displaymath} y^2={b^2x^2\over x^2-a^2}. \end{displaymath}$

(4)

In parametric form,

$\displaystyle x$	$\textstyle =$	$\displaystyle a\sec t$	(5)
$\displaystyle y$	$\textstyle =$	$\displaystyle b\csc t.$	(6)

The Curvature is

$\begin{displaymath} \kappa={3ab\csc^2t\sec^2t\over(b^2\cos^2t\csc^2t+a^2\sec^2t\tan^2t)^{3/2}}. \end{displaymath}$

(7)

References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 71, 1989.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 127 and 130-131, 1972.