Polar Coordinates

In Cartesian coordinates, you locate a point by going right and up: (x, y). In polar coordinates, you locate a point by giving a distance from the origin and an angle: (r, theta). It is like giving directions as “walk 5 steps at 40 degrees” instead of “go 3 steps right and 4 steps up.” Some curves that are nightmarish in Cartesian form become elegant and simple in polar.

Polar vs. Cartesian

The conversion formulas are:

• x = r cos(theta), y = r sin(theta) (polar to Cartesian)
• r = sqrt(x^2 + y^2), theta = arctan(y/x) (Cartesian to polar)

The gray circle has radius r. The red line goes out at angle theta. Where they intersect is the point (r, theta) in polar coordinates. The blue horizontal line shows the y-coordinate of that point.

Simple Polar Curves

Some polar equations produce beautifully simple curves:

Circles: r = a (constant radius)

The simplest polar curve is a circle of radius a centered at the origin.

Circles through the origin: r = a sin(theta)

The equation r = a sin(theta) produces a circle of diameter a, sitting on top of the origin (its center is at (0, a/2)).

In Cartesian coordinates, this becomes x^2 + (y - a/2)^2 = (a/2)^2, which we can plot:

The circle passes through the origin and has its topmost point at y = a.

Rose Curves: r = a cos(n theta)

Rose curves are among the most beautiful polar graphs. The equation r = a cos(n*theta) produces a flower-like pattern where n controls the number of petals:

• If n is odd, the rose has exactly n petals.
• If n is even, the rose has exactly 2n petals.

We can plot the rose by converting to Cartesian. For a rose curve, the x and y coordinates at angle t are x = rcos(t) = acos(nt)cos(t) and y = rsin(t) = acos(n*t)*sin(t). Here we approximate by plotting the positive and negative envelopes:

This shows r as a function of theta. The places where the curve crosses zero are where the petals begin and end. The yellow line marks the maximum petal length.

Cardioids and Limacons

The family of curves r = a + b cos(theta) produces different shapes depending on the ratio a/b:

• a/b = 1: A cardioid (heart shape)
• a/b > 1: A limacon without inner loop (a dented circle)
• a/b < 1: A limacon with inner loop

This plot shows r vs theta. When the curve dips below zero, that portion creates the inner loop of a limacon. When it stays non-negative throughout, there is no inner loop.

Spirals

The Archimedean spiral r = a*theta grows outward as the angle increases:

This shows r vs theta for the spiral. The radius increases linearly with the angle, so each revolution takes the point farther from the origin. The spacing between successive loops is constant and equal to 2pia.