Example 2.8.  The ellipse centered at the origin with a horizontal major axis of 4 units and vertical minor axis of 2 units can be represented by the parametric equation  

        ,   for  .  

Suppose we wanted to rotate the ellipse by an angle of radians and shift the center of the ellipse 2 units to the right and 1 unit up. Using complex arithmetic, we can easily generate a parametric equation r(t) that does so:

                
for  .  Figure 2.7 shows parametric plots of these ellipses.

Explore Solution 2.8.

(c) 2006 John H. Mathews, Russell W. Howell