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|
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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