Exercise 4.  Find the image of the horizontal line    under the transformation   .

Solution 4.

Answer.   The image of the horizontal line    under the transformation   

is the ellipse    (which is traced out infinitely many times).

Hint.   Use ideas found in Example 10.13.  

Short Solution.   Substitute    into equation  (10-24)     and get the ellipse   

                        traced out one time.

Hence, the image of each segment     under the transformation   

will be the complete ellipse  .

Therefore, the image of the horizontal line    under the transformation   

is the ellipse      traced out infinitely many times.

We might be done.   

Solution.   Using Equation (5-33), we write  

                    .  

The image of the horizontal segment    is the curve in the w plane given by the parametric equations  

                    ,

We rewrite them as  

                    .  

We now eliminate x from the equations by squaring and using the trigonometric identity  .  The result is the single equation

                    .

This curve is an ellipse in the w-plane that passes through the points    and   and has foci at the points .  

The parametric equations  ,    for      will trace out one complete copy of the ellipse.  

        Therefore, the image of the  horizontal line      under the transformation     

is the ellipse      traced out infinitely many times.

We are done.   

Aside.  We can look at some graphs of the mapping  .

                      The image of    under    is the ellipse  .  

                    The image of    under    is the ellipse  .  

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell