Eight Curve (Leminiscate of Gerono)


Quartic Curve >


The eight curve (also called the lemniscate of Huygens or Lemniscate of Gerono) is a quartic (degree 4) curve that gets its name because it looks like a figure eight lying on its side.

figure eight curve
The figure eight curve with a = 4.4.


The Cartesian equation for the graph is x4 = a2(x2 – y2). The constant a shrinks or stretches the figure.

The polar equation is r2 = a2cos(2θ) sec4(θ).

The parametric equations are:

  • x = a sin(t)
  • y = a sin(t) cos(t)

The curve is bounded, closed, and has one double point at the origin (0, 0). It is symmetric about the x-axis and y-axis, where (0, 0) is the center of symmetry.

History

The eight curve is sometimes called the leminiscate of Gerono, after Camille-Christophe Gerono (1799 to 1891). However, he wasn’t the first to study it. Leibniz called it “your curve in a figure eight” in a 1691 letter to Huygens (which gave rise to the second name, the leminiscate of Huygens) [1]. .

The curve has sometimes been confused with Bernoulli’s leminiscate, due to the fact that both curves look like figure eights and both are quartic curves. However, the lemniscate of Bernoulli is formed with a different equation:
(x2 + y22 = a2(x2 + y2).

liminiscates
Leminiscate of gerono (red) and Bernoulli (green).

Any leminiscate (there are many!) is shaped like a figure eight, so it can be easy to confuse them. The important thing to remember is that although they look similar, their construction and equations are all different.

References

Graph created with Desmos.
[1] Porciau, B. The Integrability of Ovals: Newton’s Lemma 28 and Its Counterexamples. Arch. Hist. Exact Sci. 55 (2001) 479–499. Springer-Verlag.

CITE THIS AS:
Stephanie Glen. "Eight Curve (Leminiscate of Gerono)" From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/eight-curve/
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