Exercise 18.  Show the following concerning the exponential map  .  

18 (d).  The image of the vertical line segment  ,  where    is half a circle.

Solution 18 (d).

See text and/or instructor's solution manual.

Solution.    .  

Substitute     into the equations    and    and obtain

                    .

Notice that      and      and   .  

This set is the parametric representation of half a circle of radius    subtended by the angles  

We are done.   

Aside.  We can let Mathematica double check our work.

                                                            The image the horizontal strip  ,  
                                                            is the half circle   .

This solution is complements of the authors.

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