Assignment 11

Polar Equations

For

Several important types of graphs have equations that are simpler in polar form than in retangular form. For example, the polar equation of a circle having radius a and centered at the origin is simply r = a. Several other types of graphs with simpler equations in polar form are displayed below.

Limacons, when k = 1

a = 1

b = 1

k = 1

a/b = 1

Cartoid

(heart-shaped)

a = 2

b = 2

k =1

a/b = 1

Cartoid

(heart-shaped)

a = 1

b = 2

k = 1

a/b < 1

Limacon with

inner loop

a = 2

b = 1

k = 1

a/b > or = 2

Convex limacon

Rose Curves, when k > 1

a = 1

b = 1

k = 2

a = 1

b = 1

k = 3

Graph has k petals, 2.
Graph has k petals, 3.

a = 1

b = 2

k = 2

a = 1

b = 2

k = 3

a/b < 1, inner loops formed

k is even, inner loops fall opposite petals

a/b < 1, inner loops formed

k is odd, inner loops are in petals

a = 2

b = 1

k = 2

a = 2

b = 1

k = 3

a/b > or = 2, convex

a/b > or = 2, convex

a = 1

b = 1

k = 2

a = 1

b = 1

k = 2

For new blue graph, 2k petals because k is even.

For new blue graph, 2k petals because k is even. Notice the change from cos to sin rotates the graph by 90/k = 45 degrees in the counterclockwise direction.

a = 1

b = 1

k = 3

a = 1

b = 1

k = 3

For new blue graph, k petals because k is odd.

For new blue graph, k petals because k is odd. Notice the change from cos to sin rotates the graph by 90/k = 30 degrees in the counterclockwise direction.

a = 1

b = 2

c = 3

Because the blue graph does not have a, a/b < 1 does not cause inner loops in it.

a = 2

b = 1

c = 3

Because the blue graph does not have a, a/b > or = 2 does not cause convexity in it.