Properties of the evolute

From (14.8),

Let us choose as independent variable the lengths of the arc on the given curve; then , , , , , are functions of . Differentiating (14.18) with respect to gives

But , , from (9.5); and , from (12.1) and (12.2).

Substituting in (14.19) and (14.20), we obtain

and

Dividing (14.22) by (14.21) gives

But = slope of tangent to the evolute at , and = slope of tangent to the given curve at the corresponding point .

Substituting the last two results in (14.23), we get

*A normal to the given curve is a tangent to its evolute.*

Again, squaring equations (14.21) and (14.22) and adding, we get

But if = length of arc of the evolute, the left-hand member of (14.24) is precisely the square of (from (10.2), where , , , ). Hence (14.24) asserts that

or

That is, the radius of curvature of the given curve increases or
decreases as fast as the arc of the evolute increases. In our
figure this means that

arc

The length of an arc of the evolute is equal to the
difference between the radii of curvature of the given curve
which are tangent to this arc at its extremities.
Thus in Example 14.4.4, we observe that if we fold ( ) over to the left on the evolute, will reach to , and we have:

*The length of one arc of the cycloid (as )
is eight times the length of the radius of the generating circle.
*

david joyner 2008-11-22