Lemniscate Envelope

Suppose we have an equilateral hyperbola and we take a point on the hyperbola. Now draw a circle that is centered at and passes through the origin. If we move 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. 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 and is 2, which is the square of the distance from the center to one focus. Click Animate below to see how it is formed.