Lemniscate

# By: Ginger Rhodes

Consider the two points (3,4) and (-5,-2).  For any point (x,y) we can write the distance equations for these as

Distance 1 =

Distance 2 =

Now, lets look at these at the graphs of these equations if the distances are equal to 2.

Notice circles with radius 2 are graphed.  The center of the blue circle is (3,4) and the center of the green circle is (-5,-2).

Suppose we look at the sum of these equations.

Lets look at the graph of the sum when it equals 15.

Now, lets look at the sum when it equals 50.

Lets consider the product of the distance equations.

To look at an animation of the product of distances equations click here.

If the two given points are (-a,0) and (a,0) then the lemniscate has its center at the origin (0,0) and major axis along the x-axis.  For example, let a=0.  Then

will be this lemniscate:

Now, we can simplify the equation.

So the equation  can be simplified to .

In general, if the foci of the lemniscate are (-a,0) and (a,0) then the equation in Cartesian coordinates is .

Lets look at a couple of examples:

Suppose we now look at the equation .

Lets try different values of b for when a is 3.

Now we will translate into an equation in polar coordinates.

Remember  and , where  .

Therefore,

So   for .