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Sinusoidal Spiral

A curve of the form

\begin{displaymath}
r^n=a^n\cos(n\theta)
\end{displaymath}

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
$-{\textstyle{1\over 2}}$ Parabola
$-{\textstyle{1\over 3}}$ Tschirnhausen Cubic
0 Logarithmic Spiral
${\textstyle{1\over 3}}$ Cayley's Sextic
${\textstyle{1\over 2}}$ Cardioid
1 Circle
2 Lemniscate


References

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, p. 184, 1972.

Lee, X. ``Sinusoid.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/Sinusoid_dir/sinusoid.html.

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.




© 1996-9 Eric W. Weisstein
1999-05-26