(559, #39) A string is wound around a circle and then unwound while being held taut. The curve traced by the point P at the end of the string is called the involute of the circle. Supposing that the circle is of radius r and the initial point of the string is (r, 0) show that the equations of the involute are

where is the angle shown below.

Solution: First, since the line from the origin to the circle has slope the line tangent to the circle has slope . Thus, by the point-slope form of the equation of the line

(1)

Second, the amount of string unwound is equal, in length, to the arc subtended by . So we have, by the distance formula,

(2)

Solving for in (1) and substituting the result in (2) we get

That is,

Taking the square root, we have

or

To determine the sign here, suppose that the minus sign is the correct one, and then differentiate with respect to :

This is a contradiction for, by the geometry of the problem, x is initially an increasing function of .

So we have

A similar argument, starting from (1) and (2) yields

The sign here can be determined by using equation (1). We have

Thus the sign is the negative one,

by: rjm