Lemniscate
The figure-eight curve where the product of distances to two foci remains constant. Bernoulli discovered it in 1694 — a special Cassini oval that happens to cross itself, and the ancestor of the infinity symbol.
r² = a² · cos(2θ) | Area = a²
About this lab
The lemniscate of Bernoulli is defined in polar coordinates by r² = a² cos(2θ). It can also be described as the locus of points P such that |PF&sub1;| · |PF&sub2;| = a²/2, where F&sub1; and F&sub2; are the two foci separated by a distance of a√2. Jakob Bernoulli published it in 1694 in Acta Eruditorum, though he described it as a figure-eight rather than connecting it to Cassini’s earlier work on ovals.
The lemniscate is a special case of the Cassini oval — the family of curves where the product of distances to two foci is constant. When that constant exactly equals (half the focal distance) squared, the oval pinches at the center into a figure eight. The enclosed area is exactly a², a result that surprised even Bernoulli. The arc length of the full curve involves an elliptic integral that resisted closed-form evaluation and inspired important work by Euler, Fagnano, and Gauss.
Gerono’s lemniscate is a different figure-eight curve defined parametrically as x = cos(t), y = sin(t)cos(t), or equivalently x&sup4; = x² − y². It looks similar but has different curvature properties. The connection between the lemniscate and the infinity symbol (∞) introduced by John Wallis in 1655 is debated — Wallis may have derived ∞ from the Roman numeral for 1000 (CI&Ovalbar;) rather than from any mathematical curve, but the visual kinship is unmistakable.