Calculus: Analytic Geometry and Calculus, with Vectors

¢00 Cones and conics

Remark: Moving coordinate systems and vectors provide the simplest way of
obtaining neat and correct answers to problems more or less like this one. The
paths traced by the pink spots P are cycloids. For future reference, we note that
if b = a, so that P is on the rolling circle and the cycloid is an ordinary cycloid,
and if we let 0 be the angle wt through which the wheel has rotated at time t,
then the coordinates of P are
x=a(O-sin 0) y=a(l-cos0).

Figure 6.592 exhibits the ordinary cycloid having these equations. The unusual

Figure 6.592

cycloids are sometimes called curt ate cycloids or prolate cycloids, and are sometimes

called trochoids.

17 Find equations of epicycloids, that is, paths traced by points on spokes

(or extended spokes) of circular wheels which roll, without slipping, outside a

fixed circular wheel. Outline of solution: As in Figure 6.593, let the fixed circle

have radius a and have its center at the origin of an x, y coordinate system.

Let the rolling circle have radius b and have center 0' which travels with it, and

let cot (not restricted to the interval 0 5 cot < 2a) be the angle which 00' makes

Figure 6.593

with the x axis at time t so that

(1) 00' = (a + b)(cos coti + sin cotj).

We suppose that co > 0, so the spokes in

the moving wheel rotate in a positive

direction. Let the point P which traces

the epicycloid be the point for which

O'P = ci when t = 0, and let S be the

spoke of the moving wheel which (ex-

tended if necessary) contains P. The

angle ¢ (psi) through which the spoke S

has turned at time t is the sum of two

angles: (i) the angle 0 through which it would turn if its wheel rolled the distance

awl along a straight line and (ii) the angle cot through which it would turn if it slid

without rolling on the fixed circle. We find that awt= bB, so 0 = (a/b)wt,

(2) a

b

bwt,