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Lemniscate

Lemniscate

A curve usually expressed in polar coordinates that resembles a figure eight.

 

Figure-eight shaped curve labeled "Lemniscate" with polar equations r²=a²cos2θ or r²=a²sin2θ shown above the curve.

 

See also

Polar curves

Key Formula

r2=a2cos⁡(2θ)r^2 = a^2 \cos(2\theta)
Where:
  • rr = Distance from the origin to a point on the curve
  • aa = Positive constant that determines the size of the loops; the farthest point from the origin is at distance $a$
  • θ\theta = Angle measured from the positive $x$-axis in polar coordinates

Worked Example

Problem: Sketch the lemniscate r2=9cos⁡(2θ)r^2 = 9\cos(2\theta) and find the angles where the curve passes through the origin.
Step 1: Identify the parameter. Here a2=9a^2 = 9, so a=3a = 3. The maximum distance from the origin is 3.
a=3a = 3
Step 2: Determine where r=0r = 0. Set cos⁡(2θ)=0\cos(2\theta) = 0, which gives 2θ=π2+nπ2\theta = \frac{\pi}{2} + n\pi for any integer nn.
θ=π4,  3π4,  5π4,  7π4\theta = \frac{\pi}{4},\; \frac{3\pi}{4},\; \frac{5\pi}{4},\; \frac{7\pi}{4}
Step 3: Determine where the curve exists. Because r2r^2 must be non-negative, you need cos⁡(2θ)≥0\cos(2\theta) \geq 0. This holds when 2θ2\theta is in [−π2,π2][-\frac{\pi}{2}, \frac{\pi}{2}] or [3π2,5π2][\frac{3\pi}{2}, \frac{5\pi}{2}], i.e., θ∈[−π4,π4]\theta \in [-\frac{\pi}{4}, \frac{\pi}{4}] and θ∈[3π4,5π4]\theta \in [\frac{3\pi}{4}, \frac{5\pi}{4}].
θ∈[−π4, π4]∪[3π4, 5π4]\theta \in \left[-\tfrac{\pi}{4},\, \tfrac{\pi}{4}\right] \cup \left[\tfrac{3\pi}{4},\, \tfrac{5\pi}{4}\right]
Step 4: Find the tip of each loop. The maximum r=3r = 3 occurs when cos⁡(2θ)=1\cos(2\theta) = 1, i.e., θ=0\theta = 0 and θ=π\theta = \pi. So the two loops extend along the positive and negative xx-axis.
rmax⁡=3 at θ=0 and θ=πr_{\max} = 3 \text{ at } \theta = 0 \text{ and } \theta = \pi
Step 5: Plot the curve. One loop lies to the right of the origin (around θ=0\theta = 0) and the other to the left (around θ=π\theta = \pi). Both loops cross through the origin at θ=±π4\theta = \pm\frac{\pi}{4} and θ=3π4,5π4\theta = \frac{3\pi}{4}, \frac{5\pi}{4}, forming a symmetric figure eight.
Answer: The lemniscate r2=9cos⁡(2θ)r^2 = 9\cos(2\theta) is a figure-eight curve centered at the origin with each loop extending 3 units along the xx-axis. It passes through the origin at θ=π4\theta = \frac{\pi}{4} and θ=3π4\theta = \frac{3\pi}{4} (and their equivalents).

Frequently Asked Questions

What is the difference between r2=a2cos⁡(2θ)r^2 = a^2 \cos(2\theta) and r2=a2sin⁡(2θ)r^2 = a^2 \sin(2\theta)?
Both produce the same figure-eight shape, but they are rotated relative to each other. The cosine version has its loops along the xx-axis (horizontal), while the sine version has its loops along the lines y=xy = x and y=−xy = -x, rotated 45°45° from the cosine form.
Why is it called a lemniscate?
The name comes from the Latin word 'lemniscatus,' meaning 'decorated with ribbons.' The curve was studied by Jacob Bernoulli in 1694 and is sometimes called the 'lemniscate of Bernoulli.' Its shape also resembles the infinity symbol ∞\infty.

Lemniscate vs. Limaçon

Both are polar curves, but they have very different shapes. A lemniscate (r2=a2cos⁡(2θ)r^2 = a^2\cos(2\theta)) always forms a figure eight that passes through the origin. A limaçon (r=b+acos⁡θr = b + a\cos\theta) can take various shapes — a cardioid, a dimpled curve, or a loop — depending on the ratio of bb to aa. A limaçon with an inner loop may look vaguely similar, but it does not cross itself at the origin the way a lemniscate does.

Why It Matters

The lemniscate is a key example when studying polar curves because it requires you to handle r2r^2 rather than just rr, which introduces the constraint that r2≥0r^2 \geq 0. It appears in calculus courses when practicing area calculations in polar coordinates — finding the area of one loop is a classic integration exercise. Beyond coursework, the lemniscate shape arises in physics (electromagnetic field lines) and in the design of racetracks and optical instruments.

Common Mistakes

Mistake: Forgetting that r2r^2 must be non-negative, and trying to plot the curve for all values of θ\theta.
Correction: Before plotting, determine where cos⁡(2θ)≥0\cos(2\theta) \geq 0 (or sin⁡(2θ)≥0\sin(2\theta) \geq 0). The curve only exists for those θ\theta-intervals. Angles where the expression is negative produce no real value of rr.
Mistake: Confusing r2=a2cos⁡(2θ)r^2 = a^2\cos(2\theta) with r=acos⁡(2θ)r = a\cos(2\theta), which is a rose curve.
Correction: The equation r=acos⁡(2θ)r = a\cos(2\theta) produces a four-petaled rose, not a figure eight. The lemniscate equation has r2r^2 on the left side, which is essential to its two-lobed shape.

Related Terms

  • Polar Coordinates — Coordinate system used to define lemniscates
  • Polar Curves — Family of curves that includes lemniscates
  • Curve — General concept a lemniscate is an example of
  • Rose Curve — Polar curve with petals, often confused with lemniscate
  • Limaçon — Another polar curve with possible loop
  • Cardioid — Heart-shaped polar curve, special limaçon case
  • Area in Polar Coordinates — Common application: finding area of a loop