Polar equations and their graphs
When the distance from the pole is a rule in the angle, what curves come out, and how can you sketch them without plotting a hundred points?
Where you are 3 inside
What this area is about
A polar equation r = f(\theta) draws a curve: for each direction \theta, walk out the distance r, where a negative r means walking |r| in the direction \theta + \pi. A point lies on the graph when at least one of its polar names satisfies the equation. Some curves that are awkward in x and y have short polar equations, and three families are worth knowing on sight.
How these fit together
A procedure to start with. Polar circles and symmetry tests covers r = a, \theta = c, r = a\cos\theta and r = a\sin\theta (with a \ne 0), and gives symmetry tests on the equation that cut the work of sketching other curves.
Two families of curves. Limacons and cardioids are r = a + b\cos\theta or r = a + b\sin\theta (a, b \ne 0); the ratio \displaystyle \frac{|a|}{|b|} decides between an inner loop, a cardioid, a dent, or a plain loop. Rose curves are r = a\cos n\theta or r = a\sin n\theta (a \ne 0, n a positive integer), whose petal count depends on whether n is odd or even.
A path through it
Circles and the symmetry tests come first, since limacons and roses are each introduced by changing a circle’s equation. Then take the two families; each points to the other where their habits overlap.
These curves are where polar coordinates pay off: each is a short equation in r and \theta and a long one in x and y.