Sinusoidal Spiral
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.
n curve
-2 hyperbola
-1 line
-1/2 parabola
-1/3 Tschirnhausen
cubic
1/3 Cayley's
sextic
1/2 cardioid
1 circle
The curvature and tangential angle are
kappa(t) = [画像:(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, SpiralExplore with Wolfram|Alpha
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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." https://mathshistory.st-andrews.ac.uk/Curves/Sinusoidal/.Referenced on Wolfram|Alpha
Sinusoidal SpiralCite this as:
Weisstein, Eric W. "Sinusoidal Spiral." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SinusoidalSpiral.html