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Cruciform

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Cruciform

A cruciform is a plane quartic curve also called the cross curve or policeman on point duty curve (Cundy and Rollett 1989). It is given by the implicit equation

x^2y^2-b^2x^2-a^2y^2=0,
(1)

which is equivalent to

(a^2)/(x^2)+(b^2)/(y^2)=1
(2)

and

y^2=(b^2x^2)/(x^2-a^2).
(3)

In parametric form,

x=asect
(4)
y=bcsct.
(5)

The curvature is given by

kappa=(3abcsc^2tsec^2t)/((b^2cos^2tcsc^2t+a^2sec^2ttan^2t)^(3/2))
(6)
=(3abcos^4tsin^4t)/((b^2cos^6t+a^2sin^6t)^(3/2)),
(7)

which, in the special case a=b, reduces to

kappa=(48sqrt(2)cos^4tsin^4t)/(a[5+3cos(4t)]^(3/2)).
(8)

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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.

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Cruciform

Cite this as:

Weisstein, Eric W. "Cruciform." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Cruciform.html

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