Calculus: Analytic Geometry and Calculus, with Vectors

402 Cones and conics

the case in which c = b and b = a/4, we obtain the hypocycloid of four
having the equation

(2) r =

4

[3 cos q5 + cos 3q]i

+

4

[3 sin q5 - sin 3q5]j

or

(3) r = a[cos3 wti + sins wtj]

or

(4) x35 + y%= a3i.

CUSPS

19 Persons interested in machinery should determine the path traced by a
cog on the inner wheel of a hypocyclic gear when the radius of the inner wheel is
just half the radius of the outer wheel.
20 A rod of length 2a, always in the xy plane, is whirling about its center with
angular speed w and, at the same time, is so thrown that its center has coordinates
(It, Bt - CO) at time t. Supposing that a red spot on the stick started at the
point (O,a) at time t = 0, find its position and velocity and acceleration at later
times. Try to solve the problem without use of the following outline. If
unsuccessful, look hastily at the outline to get some ideas and then try to solve
the problem with the outline out of sight. Outline of solution: Use vectors and
a moving coordinate system. Let an x, y coordinate system with unit vectors
i and j be drawn and remain fixed (that is, always in the same place). Let an
x', y' coordinate system have origin 0' at (17t, Bt - Ct2) at time t, and let its unit
vectors i' and j' have the directions of i and j so that, in the world of the moving
coordinate system, the stick is doing nothing but rotate about 0'. Letting r
be the vector running from 0 to P, we have

r=00'+O'P
= (alt + a cos wt)i + (Bt - Ct2 + a sin wt)j
so

and

v=(tl-awsin wt)i+(B-2Ct+awcoswt)j

a = (-awl cos wt)i + (-2C - awl sin wt)j.

21 Let a and b be positive constants, and let G be the graph in the x', y' plane
of the first of the equations

• a

Find two positive constants A and v such that G will be the graph of the second

equation in the x, y plane, the primed and unprimed coordinates being related

by the formulas x' = Ax and y' = Xy. Jnr.: X = (ira2/b) 4, o = (1/41ra2b)3'.

6.6 Quadric surfaces This brief section, which contains no terminal

list of problems, can be omitted, but it can be read even by students who