# Lemniscate of Gerono

## History

Studied by Camille-Christophe Gerono (1799 – 1891).

## Description

Lemniscate of Gerono is also known as the figure eight curve. It can be constructed as follows: Let there be a circle of radius 1 centered on the Origin. Let P be a point on the circle. Let M be the intersection of x==1 and a horizontal line passing P. Let Q be the intersection of line OM and a vertical line passing P. The trace of Q as P moves on the circle is the Figure Eight curve.

## Formula

- Parametric: {1,Sin[t]}*Cos[t], Period: 2 π.
- Cartesian: x^4 == x^2 - y^2

The parametric formula is directly derived. Starting with P:={Cos[t],Sin[t]}. The vertical line passing P would be x==Cos[t]. The point M would be {1,Sin[t]}. Solving for the insersection of line OM and x==Cos[t] results the formula.

For the Cartesian formula, eliminate the trig terms in x==Cos[t], y==Cos[t]*Sin[t] by squaring both side and add the two equations and use the identity Cos[t]^2+Sin[t]^2==1.

## Properties

### Second Construction

Let there be a circle centered on {1/2,0} of radius 1/2. Let N be the intersection of the circle and line OP. The intersection Q of vertical line thru P and horizontal line thru N traces the curve.

## Related Web Sites

See: Websites on Plane Curves, Plane Curves Books.

The Construction property is from http://www.mathcurve.com/courbes2d/gerono/gerono.shtml by Robert Ferreol, Jacques Mandonnet.

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