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.

The Cartesian equation for the graph is x

^{4}= a

^{2}(x

^{2}– y

^{2}). The constant

*a*shrinks or stretches the figure.

The polar equation is r^{2} = a^{2}cos(2θ) sec^{4}(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:

(x^{2} + y^{22} = a^{2}(x^{2} + y^{2}).

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.