Sinusoidal Spirals

Polar equation:
r^p = a^p cos(p)

Click below to see one of the Associated curves.

Definitions of the Associated curves	Evolute
Involute 1	Involute 2
Inverse curve wrt origin	Inverse wrt another circle
Pedal curve wrt origin	Pedal wrt another point
Negative pedal curve wrt origin	Negative pedal wrt another point
Caustic wrt horizontal rays	Caustic curve wrt another point

If p = -1 we have a line.

If p = 1 we have a circle.

If p = ¹/₂ we have a cardioid.

If p = ^-1/₂ we have a parabola.

If p = -2 we have a hyperbola.

If p = 2 we have a lemniscate of Bernoulli.

Sinusoidal spirals were first studied by Maclaurin.
They are not, of course, true spirals.

The pedal curve of sinusoidal spirals, when the pedal point is the pole, is another sinusoidal spiral.

The sinusoidal spiral r^p = a^p cos(p) inverts to r^p = a^p/cos(p) if the centre of inversion is taken at the pole.

The URL of this page is:
http://www-history.mcs.st-andrews.ac.uk/history/Curves/Sinusoidal.html