Sinusoidal Spiral

A curve of the form

$$r^n=a^n\cos(n\theta)$$

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.