Sinusoidal spiral

In algebraic geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates

r^{n}=a^{n}\cos(n\theta )\,

where $a$ is a nonzero constant and $n$ is a rational number other than 0. With a rotation about the origin, this can also be written

r^{n}=a^{n}\sin(n\theta ).\,

The term "spiral" is a misnomer, because they are not actually spirals, and often have a flower-like shape. Many well known curves are sinusoidal spirals including:

Rectangular hyperbola ( $n = - 2$ )
Line ( $n = - 1$ )
Parabola ( $n = - 1/2$ )
Tschirnhausen cubic ( $n = - 1/3$ )
Cayley's sextet ( $n = 1/3$ )
Cardioid ( $n = 1/2$ )
Circle ( $n = 1$ )
Lemniscate of Bernoulli ( $n = 2$ )

The curves were first studied by Colin Maclaurin.

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