GeoGebra Classroom

# Lemniscate-Infinity

Author:
Will Le
Lemniscates are curves which have the shape of "∞" (infinity symbol). Here, we explore a family of quartic (degree 4) lemniscates given by the equation:

• Lemniscate of Bernoulli﻿ [a=b=1 & c=d] : A natural modification of ellipse so that the product of distances to the 2 foci is a constant. It's also the quadratic complex polynomial lemniscate﻿ given by .
• Lemniscate of Booth (hippopede) [a=b=1 & cd > 0] : The intersection of a torus and a plane, where the plane is parallel to the axis of the torus and tangent to it on the interior circle. The Bernoulli's one is a special case of this. Both c & d must have the same sign which determines the lemniscate's direction (positive = horizontal, negative = vertical).
• Lemniscate of Gerono [a=b=0 & c=d=1] and its generalization [c , d > 0] : The planar projection of Viviani's curve which is the intersection of a sphere with a cylinder tangent with it (or with a cone).
• Alain's curve﻿ [a=-b=1 & cd > 0] : The planar projection of the intersection of the elliptical cone with the hyperbolic paraboloid. The sign of c, d determines direction (just like in hippopede). For , as 1 < k ≤ 2 the lemniscate morphs into a double cup, and when k ≤ 1 it splits into four open curves.
• Conics [a = b2] & [cd=0; c+d=0]: When d < c and one of them is zero and , the lemniscate becomes a double ellipse (or double circle if a=b=1). When c=-d & a=b=1, it degenerates into a single circle. When a=d=0 < c, it degenerates into a single ellipse (b > 0) or a hyperbola (b < 0). When , it becomes a double hyperbola (d=0) or a hyperbola with asymptotes (d=-cb).
• Devil's curve﻿ [a < 0]: When a is negative, the lemniscate is clamped by a pair of strings resembling the toy diabolo﻿ (Chinese yo-yo). As b, c, d vary, it morphs between horizontal and vertical lemniscates, and in the middle, it degenerates into an ellipse with a cross. Note that sometimes the devil's lemniscate is very similar to the non-devil one, but actually not the same!
Moreover, there are many other curves with the lemniscate shape. Eg.
• Watt's curve: A sextic (degree 6) polynomial curve whose Lemniscate of Bernoulli﻿ and Lemniscate of Booth are special cases.
• Cayley's ovals: Equipotential lines of two identical charges (in 3D) are octic (degree 8) polynomial curves where a lemniscate can be found in the middle. When the field is constrained in 2D, the corresponding equipotential lines are the quartic Cassini's ovals﻿ where a lemniscate of Bernoulli is in the middle.
• Analemma﻿: An geostationary satellite﻿ with some inclination from the Earth's equator will trace a lemniscate called "hippopede of Eudoxus﻿" on the sky. Let's see Satellite Trace﻿ with inclination > 0, latitude = 0, a = b.
• Meander curves: At a critical point (ϕmax≈0.766π), the river meanders close back into a lemniscate! Let's see this special clothoid﻿ with sine curvature.
• Elastic curves﻿: Very much like meander curves, there's a lemniscate elastic wire at the critical point k≈0.65222.
• Double egg curve﻿, and so on...