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Hyperbola Inverse Curve

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HyperbolaInverseCenter

For a rectangular hyperbola

x=asect
(1)
y=atant
(2)

with inversion center at the origin, the inverse curve is

x_i=(2kcost)/(a[3-cos(2t)])
(3)
y_i=(ksin(2t))/(a[3-cos(2t)]),
(4)

which is a lemniscate.

HyperbolaInverseFocus

For a rectangular hyperbola with inversion center at the focus (asqrt(2),0), the inverse curve is

x_i=asqrt(2)-(2kcost(sqrt(2)cost-1))/(a[5-4sqrt(2)cost+cos(2t)])
(5)
y_i=(ksin(2t))/(a[5-4sqrt(2)cost+cos(2t)]),
(6)

which is a limaçon.

HyperbolaInverseVertex

For a rectangular hyperbola with inversion center at the parabola vertex (a,0), the inverse curve is

x_i=a+(kcost)/(2a)
(7)
y_i=(kcostsint)/(2a(1-cost)),
(8)

which is a right strophoid.

HyperbolaInverseSq3Vertex

For a non-rectangular hyperbola with a=sqrt(3)b and inversion center at the parabola vertex, the inverse curve is

x_i=sqrt(3)(b+(kcost)/(2b(2-cost))]
(9)
y_i=(kcostcos(1/2t))/(2b(2-cost)),
(10)

which is a Maclaurin trisectrix.

See also

Hyperbola, Inverse Curve

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References

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, p. 203, 1972.

Referenced on Wolfram|Alpha

Hyperbola Inverse Curve

Cite this as:

Weisstein, Eric W. "Hyperbola Inverse Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HyperbolaInverseCurve.html

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