Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

THE STRAIGHT LINE

491

The Helix. Given the cylinder
x2 + y2 = a2
with the axis of z as axis. The circle in the (x, y)-plane on
which the cylinder is erected has the parametric representation (Ch. VII, ~ 10):
x a cos 0, y = a sin 0, = 0,
where 0 is the angle which the radius, OP, to the point
P: (x, y, 0) makes with the positive axis of x. On the ruling of
the cylinder through P mark the point P' at a height above P*
equal to a constant multiple, kO, of the
angle 0. The coordinates of P' are
(3) x a cos 0, y = a sin O, z= -c. I I
When -= 0, P and P' coincide in the
point P0 on the axis of x. As 0 increases -..-"
from 0 to 27r, the point P traces the circle,
and the point P', always directly above P,
traces a locus on the cylinder, encircling it P
just once. When 0 increases from 2w to /.
4r, P retraces the circle, whereas pr con- - Potinues on its rising path, encircling the
cylinder a second time. Consequently,
when 0 increases through all positive values,
the locus traced by P' encircles the cylinder
infinitely many times.
As 0 decreases from zero through all FoG. 8
negative values, P', starting from Po, encircles the lower half of the cylinder infinitely many times.
The complete locus of P' is, then, an unbroken curve continuously winding about the cylinder in both directions. It is this
curve which is represented parametrically by the equations (3).
Since the height of P' above (or below) P is always proportional to the angle 0 through which the radius OP of the circle
has turned, the curve (3) which P' traces is mounting on the
* Above P, if 0 is positive; below P, if 0 is negative.