Exercise 4.  Explain why the diagrams in Figures 5.7 and 5.8 came out they way they did.  

Solution 4 (a).

See text and/or instructor's solution manual.

Solution 4 (a).   Consider Figure 5.7, start with equation (5-34)  .   

For  ,  and  ,  consider the line segment  .  The image is

                      

But  ,  

setting    and  ,  

we see that    is a portion of the hyperbola  .

                    Consider the mapping  .

                    The image of    is a portion of the hyperbola  .

We are done.   

        Similarly,  for the other values      and     

the image of    will be a portion of a  hyperbola.

                    Consider the mapping  .

                    The image of    will be a portion of a  hyperbola.

In Section 10.4, we will give a more detailed analysis of the mapping  .

This solution is complements of the authors.

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