Exercise 5.  Find the image of the rectangle    under the transformation   .  

Solution 5.

Answer.   The image of the rectangle    under the transformation    is

                    ,  

which is the region in the first quadrant    that lies

inside the ellipse    and to the left of the right branch of the hyperbola  .  

Hint.   Extend the results in Example 10.13.  You can use the information in Figure 10.17.

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

                    .  

If   ,   then the image of the vertical line      is the curve in the w plane given by the parametric equations  

                      and  ,  for  .  

The inequalities  ,  and    imply that the image points lie in the first quadrant .  

Eliminate y from the parametric equations and the result is  .  

When the image curve approaches the positive v axis where  .  

When the image curve approaches the right branch of the hyperbola    .  

        Hence, the image of the semi-infinite strip      under the mapping     is  

the portion of the first quadrant    that lies to the left of the right branch of the hyperbola  .  

        Next, using Equation (5-33), we write  

                    .  

        If  ,  then the image of the horizontal segment    

is the curve in the w plane given by the parametric equations  

                    ,

and can be rewritten 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 inequality      implies that      and   ,   and we can conclude

that the ellipse     lies inside the bounding ellipse   .  

        Hence, the image of the   ,   lies inside the bounding ellipse   .  

        Therefore, the image of the rectangle    under the transformation    is  

                    ,  

which is the region in the first quadrant    that lies

inside the ellipse    and to the left of the right branch of the hyperbola    .  

We are done.   

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

                                        The mapping   .

                                        Observe the points and their images    

                                                  

This solution is complements of the authors.

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