Calculus: Analytic Geometry and Calculus, with Vectors

546 Polar, cylindrical, and spherical coordinates

which may help us determine x(t) and y(t). Using the fact that the distance
from P to Q is always b (the radius of the wheel), obtain the formula

t girt 2

(4) [x(t) - a cos

2Yrr, ]2

+ [y(t) - a sin 7 ] = b2 sin 2

2irat

FT-

Show that

(5) [x(t)]2 + [y(t)]2 = a2 + b2 Sin2b7at

With or without assistance from these formulas, derive the formulas

2

x(t) = a cos Tt + b sin Tt sin bTt

y(t) = a sin 2Tt - b cos

2Ttsin -y-

z(t) = b - b cos

27rat

bT

and use them to find the velocity and acceleration of P when t = 0. Remark:

Mechanisms involving rolling wheels (or gears) appear in machinery in various

ways, and we have the preliminary idea that we can start studying these things.

10 Let a be a positive number. The point P lies on a line through the origin

which intersects the line having the rectangular equation x = a at a point Q,

and IPQI is equal to the distance from Q to the x axis. The set of such points P

is a strophoid. Find the polar equation of the strophoid. -4ns.:

p2 cos 0- 2ap+ a2 cos 0 = 0.

11 Supposing that a > 0, prove that the line having the rectangular equation

y = a is an asymptote of the hyperbolic spiral having the polar equation p4) = a.

12 Supposing that a > 0, prove that the x axis is an asymptote of the lituus

having the polar equation p2¢ = a2.

13 Show that transforming the first of the equations

(1) p= 4a cos 0- a sec 0, x4+xy2+ay2-3ax2=0

from polar to rectangular equations gives the second. Remark: The graph of

these equations is a trisectrix of Maclaurin. It is possible to use a formula for

tan 30 to show that if 0 is the origin, if Q is the point (2a, 0), if P is a point on the

trisectrix in the first quadrant, if 6 is the angle in the interval 0 < 0 < ir which

the line OQ makes with the positive x axis, and if ¢ is the acute angle which OP

makes with the positive x axis, then 6 = 30. Thus the trisectrix provides a

method (but not a ruler-and-compass method) for trisecting angles. Particu-

larly when 0 < 0 < 7r/2, the number p in (1) has an elegant geometric interpreta-

tion. It is JOBI - 10-41 where 14 and B are the points wherea line through 0

meets the line having the rectangular equation x = a and the circle of radius

2a having center Q.

14 The innocent graph of the equation

1

Y=1,+x2