Involute of a Circle

Parametric Cartesian equation:
x = a(cos(t) + t sin(t)), y = a(sin(t) - t cos(t))


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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The involute of a circle is the path traced out by a point on a straight line that rolls around a circle.

It was studied by Huygens when he was considering clocks without pendulums that might be used on ships at sea. He used the involute of a circle in his first pendulum clock in an attempt to force the pendulum to swing in the path of a cycloid.

Finding a clock which would keep accurate time at sea was a major problem and many years were spent looking for a solution. The problem was of vital importance since if GMT was known from a clock then, since local time could be easily computed from the Sun, longitude could be easily computed.

The pedal of the involute of a circle, with the centre as pedal point, is a Spiral of Archimedes.

Of course the evolute of an involute of a circle is a circle.


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JOC/EFR/BS January 1997

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