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 = [画像:(2kcost)/(a[3-cos(2t)])]

(3)

y_i = [画像:(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 = [画像:asqrt(2)-(2kcost(sqrt(2)cost-1))/(a[5-4sqrt(2)cost+cos(2t)])]

(5)

y_i = [画像:(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 = [画像:a+(kcost)/(2a)]

(7)

y_i = [画像:(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 = [画像:sqrt(3)(b+(kcost)/(2b(2-cost))]]

(9)

y_i = [画像:(kcostcos(1/2t))/(2b(2-cost)),]

(10)

which is a Maclaurin trisectrix.

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References

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

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Hyperbola Inverse Curve

Cite this as:

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

Hyperbola Inverse Curve

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