Exercise 6.  Consider the mapping  .  

If  ,  show that the lines    are mapped onto orthogonal parabolas.

Solution 6.

See text and/or instructor's solution manual.

Solution Method I.   Applying Theorem 10.1,  for   ,   we have      and since  ,   

it follows that    .   Hence    is conformal at  .  

Since   is conformal at  ,  the lines    and    

which intersect orthogonally at the point  ,  will be mapped onto curves

which intersect orthogonally at the point  .  

Solution Method II.   In Example 2.12 in Section 2.2, we saw that the image of the line is the parabola    and  .  

At the point    the slope of this parabola is    and we can write it in the   form

                    .  

Also, the image of the line    is the parabola    and  .  

At the point    the slope of this parabola is    and we can write it in the   form

                    .  

Since  ,  the curves are orthogonal.

Solution Method III.   For the mapping  ,  the lines  

                       and     

intersect orthogonally at the point  .

Their image curves are

                    ,   and  


                    ,

and their tangent vectors are  

                    ,   and

                    .

At the point    the tangent vectors to the curves   are

  and  ,  respectively,  and we have

                      

Therefore, the lines    are mapped onto orthogonal parabolas.

We are done.   

Aside.  We can let Mathematica illustrate our work.

                                   The transformation      

                                   The transformation      

We are really done.

Caveat.  At a point    where     we will have

                    .

Applying Theorem 10.2 we see that the mapping   magnifies angles at the vertex    by the factor  .

                      For     at    we have      and   ,  and
                      the mapping   magnifies angles at the vertex    by the factor  .

Remark.  In Section 2.2 we introduced the mapping  .

This solution is complements of the authors.

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