Lemniscate of Bernoulli

The Lemniscate of Bernoulli is a fascinating plane curve discovered by the Swiss mathematician Jakob Bernoulli in 1694. Its characteristic figure-eight shape makes it one of the most recognizable curves in mathematics. The curve can be defined as the set of points for which the product of the distances to two fixed points (foci) remains constant.

Beyond its visual appeal, the lemniscate appears in geometry, complex analysis, and dynamical systems. It is often compared to the infinity symbol, although the two shapes are not exactly the same. Exploring the curve provides insight into polar coordinates, algebraic curves, and mathematical visualization.

The interactive Desmos model below allows you to examine the Lemniscate of Bernoulli in 3D and explore its structure dynamically.

In polar coordinates, the lemniscate is commonly represented by the equation:

$$ r^2 = a^2 \cos(2\theta) $$

where $a$ determines the size of the curve. As $\theta$ varies, the equation traces the two symmetric loops that form the characteristic figure-eight shape.

Try interacting with the visualization to observe how the curve behaves from different perspectives and to gain a deeper understanding of its geometry.