Fermat's Spiral

Polar equation:
r2 = a2θ


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 your browser can handle JAVA code, click HERE to experiment interactively with this curve and its associated curves.

This spiral was discussed by Fermat in 1636.

For any given positive value of θ there are two corresponding values of r, one being the negative of the other. The resulting spiral will therefore be symmetrical about the line y = -x as can be seen from the curve displayed above.

The inverse of Fermat's Spiral, when the pole is taken as the centre of inversion, is the spiral r2 = a2/θ.

For technical reasons with the plotting routines, when evolutes, involutes, inverses and pedals are drawn only one of the two branches of the spiral are drawn.


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JOC/EFR/BS January 1997

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