 EMAT 6680 Assignment 11

Polar Equations

Melissa Bauers Summer 2002

This assignment is about polar equations. We will examine the equation of Let us first examine the effects of theta . We will keep a, b, and k constant.

CASE #1

a = 0

b and k = 1 CASE #2

Now we will vary k.

k = 2 k = 3 k = 4 k = 5 As the value of k changes the number of leafs on the graph change. Click on link for a dramatic animation of k ranging from -25 to 25. Notice in the animations the negative numbers give the same picture as the positive.

N-Leaf Rose Animation

CASE #3

Vary the values of b.

a = 0

k = 1

Let us start with b = 2. We will overlay onto the graph when b = 1. b = 5 Notice changing b increases the size of the leafs. Click here for an animation of b going from -4 to 4. Notice in the animation the negative numbers give the same picture as the positive.

CASE #4

We will go back to k =4, b = 1, and a will vary.

a =1 a = 2 a = 3 When a varies the "leaf rose" formation changes. Click below for an animation of a changing from -8 to 8. Notice the negative numbers look the same as the positive numbers.

Variation of a

SPECIAL CASE

When a and b are the same the graphs are called cardiods. If a does not equal b, then graph is called a limacon. Here are some examples:

a = 1

b = 2

k = 1 a = 1

b = -2 a = 3

b = 2 As the difference of a and b increases when a > b and b does not equal to 0, the graph resembles a circle. As the difference of a and b increases when b > a and a does not equal to 0, then the inner circle becomes bigger.