Calculus: Analytic Geometry and Calculus, with Vectors

6.5 Translation and rotation of axes 401

and

(3) O'P = c

Icosa b b

wti + sin a b b wtj 1

at time t. From (1) and (3) we obtain

(4) r = (a b) cos wt -f c cos +b wt i

[

ab

+[(a+b)sinwt+csinabbwt,j

for the displacement vector of the point P on the epicycloid at time t. An ordi-
nary epicycloid is obtained by setting c = -b so P starts at the initial point of
tangency of the two circles. Remark: For the case in which c = -b and b = a,
we obtain the epicycloid of one cusp having the equation

(5) r = a[2 cos wt - cos 2wt]i + a[2 sin wt - sin 2wt]j.

Using the trigonometric identities

(6) cos 20=cos20-sin20=2cos20-1, sin 20=2sin0cos0

enables us to put (5) in the form

(7) r - ai = 2a(1 - cos cot) (cos cod + sin wtj)

from which we see that

(8) Ir - all = 2a(1 - cos cot).

Letting p denote the distance from the point (a,0) to the point P on our epicycloid
and setting ¢ = wt puts (8) in the form

(9) p = 2a(1 - cos 0).

The polar coordinate graph of (9) is a cardioid.Thus we have discovered that
an epicycloid with one cusp is a cardioid, and we have started learning about
epicyclic gears. That the Greek prefixes epi and hypo mean outside (or above)
and inside (or below) can be remembered by those who have hypodermics put
under their skins.

(^18) Find equations of hypocycloids, that is, paths traced by points on spokes
(or extended spokes) of circular wheels which roll, without slipping, inside a
fixed larger circular wheel. Outline of solution: This problem is much like Prob-
lem 17. To get the answer from (4) of Problem 17, replace b by -b because
IOO'I = a - b. For hypocycloids the spokes of the inner wheels run backwards.
The equation giving displacement vectors of points on hypocycloids is
(1) r =[ (a- b) cos wt + c cosabbwt, is
+[(a-b)sinwt-csina t] j.
An ordinary hypocycloid is obtained by setting c = b so P starts at the initial point
of tangency of the two circles. A graph appears in Figure 7.291. Remark: For