Calculus: Analytic Geometry and Calculus, with Vectors

10.1 Geometry of coordinate systems 531

0 to a as 20 increases from 2Tr to 57r/2 and hence as ¢ increases from 7r to

Sir/4. This gives the first half of the third leaf. Three more increases

in 20 and .0 complete the third and fourth leaves as .0 increases to 2ir.

Increasing 4, beyond 2ir yields more curve but no more graph, since the

graph is retraced. The full graph is shown in Figure 10.151. In the

good old days, perhaps before clover was invented and when roses were

primitive, someone called this graph the rose with four leaves.

Figure 10.151

The polar graph of the equation

(10.16) p=a(l+cos0)

Figure 10.152

is obtained much more easily. As 0 increases from 0 to 7r and then to

27r, p decreases from 2a to 0 and then increases to 2a. This graph, which

is called a cardioid, is shown in Figure 10.152.

Let a be a positive constant. When 0 < 0 < 7r/2, it follows from

elementary geometry and trigonometry that the point P having polar

coordinates (p,¢) for which

(10.161) p=acos4

lies on the circle of Figure 10.162. Consideration of other angles shows

that the circle is the complete graph (which, of course,

Figure 10.162

means the graph) of the equation.

A bit of novelty appears when we undertake to P

sketch a graph of the equation a

(in (^1) PZ =a 2 cos 2 ¢. (^006)
As 20 increases from -7r/2 to 0 and then to 7r/2, and

hence as ¢ increases from -ir/4 to 0 and then to