Suppose we have an equilateral hyperbola [math]x^2-y^2=1[/math] and we take a point [math]P[/math] on the hyperbola. Now draw a circle that is centered at [math]P[/math] and passes through the origin. If we move [math]P[/math] and trace all of the different circles that are made, slowly the outline of a lemniscate will take shape. We say that the lemniscate is the envelope of the circles that were formed.[br][br]The foci of the hyperbola are also the foci of the lemniscate. This particular lemniscate can be defined as the locus of all points, the product of whose distances from [math]F_1[/math] and [math]F_2[/math] is 2, which is the square of the distance from the center to one focus.[br][br]Click Animate below to see how it is formed.