Sinusoidal Spirals

Polar equation:
rp = ap cos()


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


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Sinusoidal spirals can have any rational number p in the formula above. Many standard curves occur as sinusoidal spirals.

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 rp = ap cos() inverts to rp = ap/cos() if the centre of inversion is taken at the pole.


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JOC/EFR/BS January 1997

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http://www-history.mcs.st-andrews.ac.uk/Curves/Sinusoidal.html