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

\end{displaymath} (1)

which is equivalent to
1-{a^2\over x^2}-{b^2\over y^2}=0
\end{displaymath} (2)

{a^2\over x^2}+{b^2\over y^2}=1,
\end{displaymath} (3)

or, rewriting,
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
\end{displaymath} (7)


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.

© 1996-9 Eric W. Weisstein