Assignment 11
Polar Equations
by Emily Bradley
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 (heartshaped) 


a = 2 b = 2 k =1 
a/b = 1 Cartoid (heartshaped) 

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. 