ASSIGNMENT 11

POLAR EQUATIONS

BY :  SHARREN M. THOMAS

THE PROBLEM:

Investigate r = a + b cos (kθ)

Below see the graphs, when a = b and k = 1 these heart-shaped graphs are called cardioids.

r = 2 + 2 cos θ; r = 4 + 4 cos θ; r = 5 - 5 cos θ; r = 3 - 3 cos θ

If a and b are not zero, then the graph r = a + b cos (kθ) where k = 1 are limacons.  When a/b < 1 the graph will be a limacon with an inner loop.  See below.

r = 2 + 4 cos θ; r = 2 - 3 cos θ

When 1 < a/b < 2, the graph will be a limacon with a dimple.  When a/b ≥ 2 then the graph will be a convex limacon.  See below.

r = 6 + 4 cos θ; r = 8 + 4 cos θ.

r = a cos(kθ) and r = a sin(kθ) are the general for of polar equations whose graphs are four-leafed roses.  For any positive integer n greater that 1 and any nonzero real number a has a graph that consists of a number of loops through the origin.  If n is even, there are 2n loops, and if n is odd, there are n loops.  See samples below and also press click to manipulate with graphing calculator.

r=a+b cos (kΘ) where a=b and k is a noninteger

One notices from the graphs of r=4+4 cos (1/2(Θ)),  r=4+4 cos (4/3(Θ) and r=4+4 cos (11/4(Θ)) noninteger values of k, partial leaves occur though.

r=a+b cos (kΘ) where a<b and k is an odd integer smaller leaves appear inside the larger ones.