•GRAPH POLAR EQUATIONS

10.7 Polar Graphs

Where Are Polar Coordinates and Graphs Used in the Real World?

Radar measures the things it is tracking by the angle and distance from the antenna. This maritime navigation radar shows the ship's position in the center of the display and other ships and land as distance and direction from the ship. (The straight lines attached to the other ships are computerized course projections for those ships.)

Where Are Polar Coordinates and Graphs Used in the Real World?

The pickup patterns of microphones are shown on polar plots. This plot shows the pattern for one of the most popular directional microphones used for vocal performance. The pattern is called "cardioid," (meaning heart-shaped). You can see that the shape changes a bit depending on the sound's frequency. Each circle on the plot represents a bit less than doubling of the sound reception, so this microphone "hears" things about 10 times better from the front than from the back.

Graphing Circles

a

a

cosr a sinr a

Graphing Limacons

sinr a b cosr a b

sinr a b cosr a b

Inner Loop Heart Dimple Flat

1ab

1ab

1 2ab

2ab

Graphing Rose Curvessinr a n cosr a n

Even = 2 Petalsn

sin 2r a sin 3r a sin 4r a sin 5r a

Odd = Petalsn

a a a a

Limaconsr = a b cos tr = a b sin t

a, b > 0

Rose curves

r = a cos (nt) r = a sin (nt)

n petals if n is odd2n petals if n is even

n ≥ 2

Circles and Lemniscates

r = a cos tr2 = a2 sin (2t)

a/b < 1 inner loop

a/b = 1 heart shaped

1 < a/b < 2 dimpled

a/b >= 2 convex