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Sinusoidal Spiral

A curve of the form


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


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

Lee, X. ``Sinusoid.''

Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 175, 1967.

MacTutor History of Mathematics Archive. ``Sinusoidal Spirals.''

© 1996-9 Eric W. Weisstein