lemniscats

The lemniscate is a figure-eight shaped curve. The most common lemniscate is the Lemniscate of Bernoulli, which was discovered by Jacob Bernoulli in 1694. It is defined as the locus of points such that the product of the distances from two fixed points (called foci) is constant. For the Lemniscate of Bernoulli, these foci are located at (a, 0) and (-a, 0), and the constant product is a^2. The equation of the Lemniscate of Bernoulli in Cartesian coordinates is (x^2 + y^2)^2 = 2a^2(x^2 - y^2). In polar coordinates, its equation is r^2 = 2a^2 cos(2θ).

The lemniscate has a characteristic shape resembling an infinity symbol. It is symmetric with respect to both the x-axis and the y-axis. The curve passes through the origin. The points where the curve is furthest from the origin are at (a, 0) and (-a, 0). The curve has two loops that meet at the origin.

The Lemniscate of Bernoulli has several interesting mathematical properties. For example, its arc length can be calculated using elliptic integrals. It also appears in various areas of mathematics, including geometry, calculus, and differential equations. Another notable lemniscate is the Lemniscate of Gerono, which is a different shape that also resembles a figure-eight. This curve is defined by the equation x^4 = a^2(x^2 - y^2). While similar in appearance, the mathematical properties and defining equations of the Lemniscate of Bernoulli and the Lemniscate of Gerono differ.