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Sinusoidal Spiral

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SinusoidalSpiral

A sinusoidal spiral is a curve of the form

r^n=a^ncos(ntheta),
(1)

with n rational, which is not a true spiral. Sinusoidal spirals were first studied by Maclaurin. Special cases are given in the following table.

ncurve
-2hyperbola
-1line
-1/2parabola
-1/3Tschirnhausen cubic
1/3Cayley's sextic
1/2cardioid
1circle
2lemniscate

The curvature and tangential angle are

kappa(t)=(n+1)/(acos^(1/n-1)(nt))
(2)
phi(t)=-tan^(-1)[cot(nt)].
(3)

See also

Sinusoidal Spiral Inverse Curve, Sinusoidal Spiral Pedal Curve, Spiral

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References

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, p. 184, 1972.Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 175, 1967.MacTutor History of Mathematics Archive. "Sinusoidal Spirals." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Sinusoidal.html.

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Sinusoidal Spiral

Cite this as:

Weisstein, Eric W. "Sinusoidal Spiral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SinusoidalSpiral.html

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