How to graph lemniscates in polar coordinates

A lemniscate is a polar curve with two loops that meet at the pole, giving it a figure-eight shape. The sine or cosine function and the leading sign tell us where those loops sit. We’ll use those features to sketch the curve, then check which angles give real values of r.

The four standard lemniscate equations

With a > 0, the standard forms are $r^{2} = a^{2} \cos (2 θ)$ , $r^{2} = - a^{2} \cos (2 θ)$ , $r^{2} = a^{2} \sin (2 θ)$ , and $r^{2} = - a^{2} \sin (2 θ)$ . In each case, the farthest points on the loops have distance a from the pole.

Lemniscate orientation, with a > 0
Equation	Where the loops lie
$r^{2} = a^{2} \cos (2 θ)$	Along the horizontal axis
$r^{2} = - a^{2} \cos (2 θ)$	Along the vertical axis
$r^{2} = a^{2} \sin (2 θ)$	In quadrants I and III
$r^{2} = - a^{2} \sin (2 θ)$	In quadrants II and IV

The coefficient in front of the trig function is a², so taking its positive square root gives the maximum radius. If that coefficient is 9, the tip radius is 3.

Example: Graph r² = 3 cos(2θ)

This is a positive cosine lemniscate, so the two loops lie along the horizontal axis. Here a² = 3, which gives $a = \sqrt{3}$ , or approximately 1.73.

The two horizontal loop tips are each √3 units from the pole.

At θ = 0, cos(2θ) = 1, so the right loop reaches $r = \sqrt{3}$ . At θ = π, the cosine is 1 again, and the positive radius places the tip on the negative x-axis.

The curve reaches the pole when r = 0. Solve cos(2θ) = 0 to find those angles:

θ = \frac{π}{4} + \frac{k π}{2}

Here k is any integer. Between 0 and 2π, the pole angles are π/4, 3π/4, 5π/4, and 7π/4.

Which angles give real radii?

Because r² is nonnegative for a real radius, the right side of the polar equation must also be nonnegative. For this cosine example, that means cos(2θ) ≥ 0.

On 0 ≤ θ ≤ 2π, the valid intervals are [0, π/4], [3π/4, 5π/4], and [7π/4, 2π]. At θ = π/2, the right side is −3, so there is no real radius at that angle. We leave that angle out of the plot.

Using r ≥ 0 on the valid intervals traces the whole curve. Negative radii are allowed in polar coordinates too, but they are not needed for this sketch.

Example: Graph r² = 4 sin(2θ)

This is a positive sine lemniscate, so its loops lie in quadrants I and III. Since a² = 4, its maximum radius is a = 2.

The diagonal loops lie in quadrants I and III. Each loop tip is 2 units from the pole.

The sine equals 1 when θ = π/4 and θ = 5π/4, so those are the directions of the two loop tips. At θ = 0, π/2, π, and 3π/2, the sine is 0 and the curve reaches the pole.

Within one full turn, sin(2θ) ≥ 0 on [0, π/2] and [π, 3π/2], with 2π giving the pole again. These are the angle ranges we use when plotting with a nonnegative radius.

A quick plan for sketching a lemniscate

Identify sine or cosine and the leading sign to locate the two loops.
Take the positive square root of the coefficient magnitude to find the tip radius.
Find the angles where the right side is 0 and where it reaches its positive maximum.
Sketch the two loops symmetrically about the pole, using only angles that give real radii.

Learn mathKrista KingMay 8, 2017graphs, graphing polar curves, curves, lemniscates