The Involute of a Circle

The Involute of a Circle

Introduction

The involute of a circle is the path traced out by a point on a straight line that rolls around a circle. It’s Cartesian coordinates are given by x = a(cos(theta) + t sin(theta)), y = a(sin(theta) - t cos(theta)) where a = the radius of the circle, and theta is the angle. Christian Huygens, a scientist who was working on the pendulum clock, first described it in his book Horologium Oscillatorium.

Christian Huygens

Huygens was a Dutch scientist who lived in the 1600s. He worked on many problems, including lens grinding and telescope construction, the wave theory of light, and believed that the speed of light was constant. Using lenses he developed, he discovered the first moon of Saturn in 1655. The next year he also determined the correct shape of Saturn’s rings. Since he needed accurate time for astronomy problems, he started to work on building a better clock. Eventually he patented the pendulum clock, in 1656. In Horologium Oscillatorium, he described his theories, and also defined the terms evolute and involute. The evolute is the opposite of the involute. He used the involute of a circle to make a pendulum swing in a cycloidal motion, for use in clocks at sea.

An example of a cycloid:

Polar Coordinates

To convert from Cartesian to Polar coordinates, set x= r cos theta, y= r sin theta.

r cos theta = a(cos(theta) + t sin(theta)), r sin theta = a(sin(theta) - t cos(theta))

r=a + a*t*tan(theta), r=a - a*t*cot(theta)

Properties

The involute is very useful in gears. Leonhard Euler was the first to propose to use the involute of the circle for the shape of the teeth of a gear. Gears with involute profiles provide a constant ratio of rotational speed. With other types of gears, the point of contact changes as the gear rotates. It starts close to one gear, and as it gets closer to the other, less force is applied. This makes the gear ratio change, which would make it speed up and slow down constantly. The involute gear‘s shape compensates for this, because as the contact point moves, the force applied changes, so the speed is kept constant.

References

JOC/EFR/BS January 1997

HowStuffWorks, Inc. 1998-2008

Taken from A Short Account of the History of Mathematics by W. W. Rouse Ball (4th Edition, 1908).

Shop Reference for Students and ApprenticesBy Edward G. Hoffman, Christopher J.

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