Assignment 1

# Lemniscates

Click on any equation below to see its corresponding graph. You may click the animate button on the bottom of the graph to see various values for a, b or c.

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

Note: The distance formulas above do not have corresponding graphs.

There are many graphs used to demonstrate these two distance equations. For example,

a. Consider when each is set to a non-zero constant.

Note: The movie shows the changes in the graph as c changes.

b. Consider the sum for various values of C.

c. Consider the product for various values of C.

d. 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 the following equation will be a lemniscate.

We can see by the following algebra steps, that 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

Note: The movie shows the changes in the graph as a changes.

Now consider the following equation.

Click the corresponding letter for different values of A or B.

Now we will look into polar coordinates. We can see by the following algebra steps that the equation can be translated to the following equation in polar coordinates.

or