Calculus: Analytic Geometry and Calculus, with Vectors

10.1 Geometry of coordinate systems 535

and the formula (7.389) for curvature in rectangular coordinates, show that if
p and ¢ are functions having two continuous derivatives and if the point P(t)
having polar coordinates p(t), 4(t) traverses the curve C as t increases, then the
curvature K of C at P(t) is

K = [p(t)12[4)'(t)l3 + P(t)P'(t)4)"(t) - P(t)P"(t)4)'(t) + 2[P'(t)]24)'(t)

{[P(t)4)'(t)l2 + [P'(t)]2}3i2

provided the denominator is not zero.

14 Show that letting ¢(t) = t and replacing t by 0 in the last formula of the

preceding problem gives the much simpler formula

K = [P(4))12 + 2[p'(4))]2 - P(O)P"(O)

{ [P(4))12 + [P'(4))]2} 41

for the curvature of the graph of the polar equation p = p(4)) oriented in the

direction of increasing ¢.

15 The two equations

(1)

(2)

x2 -+Y2 + IxI + IYI = a

x2+y2+x+y = a,

in which a is a given positive constant, have respective graphs G, and G2. Show

that the polar coordinate equations of these

graphs are IY I i 1

()^3

and

(4)

or

P a

1 + Isin q5I + Icos q6I

P 1+sin0+cos4)

=

a

P

1+1/2sin (4)-{ 4)

with p > 0. Obtain more or less complete

information about G, and G2-

16 The cissoid of Diodes is the set of

points P(x,y) obtained in the following way.

Let a > 0. As in Figure 10.192, let C be the

circle with center at (a,0) having radius a.

Let 0 < x < 2a. Let Q, and Q2 be points on

the upper part of C having x coordinates x

and 2a - x. Then P(x,y) is the intersection

of the line OQ2 and the vertical line through

QI. Letting ¢ be the angle which OQ2

makes with the x axis and with the line QZQ,, we see that

and, when x 3-4- a,

Y=

Figure 10.192

a when x = a

(1) tan0=y=

=y - a2- (x- a)2

x IQZQ,I 2(x - a)