# 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
. It's also the quadratic complex polynomial lemniscate given by .**product**of distances to the 2 foci is a constant - Lemniscate of Booth (hippopede) [a=b=1 & cd > 0] : The
*intersection of a*, 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).**torus**and a plane - 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 = b
^{2}] & [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(d=-cb).**hyperbola**with asymptotes - 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. Note that sometimes the devil's lemniscate is very similar to the non-devil one, but actually not the same!**ellipse**with a cross

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 (ϕ
_{max}≈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...