Solution 6.

See text and/or instructor's solution manual.

Answer.   Let    be fixed, then the circle      in the z-plane

is mapped onto the cardioid      in the w-plane.  

Solution.   The equation of a circle in the -plane with center is given in polar coordinates    by the formula  

                    .  

For the mapping    we have  ,  and  .  

Use the substitutions    and   in    to obtain  

                    .

Square both sides and get

                    .  

Interesting Fact.   

        To obtain the standard form of a cardioid use the trigonometric identity  ,  and write

                      
                              
Now make the substitution   and get the standard form of a cardioid:

                    .  

        Furthermore, the transformation    maps the point      onto the point   .

For the function     we have      and   .  

Since   ,   the mapping is not conformal at  ,  but  .  

Hence, by Theorem 10.2 in Section 10.1 the mapping    magnifies angles at    by the factor  .

        The rays tangent to the circle at    make an angle    and  ,  respectively.

The image rays tangent to the two curves forming the cusp at    make the angles    and  ,  respectively.

Since these angles point in the same direction, the cusp in the cardioid forms an angle of  0°.     

We are done.   

Aside.  We can let Mathematica double check our work.

Aside.  We can look at the equations.  

We are really done.   

Aside.  We can explore some graphs.

                    The circle      with center  .

                    Rays tangent to the circle at the origin make an angle    and  .

                    Remark.  The image of these two rays will be a single ray making an angle  .

                    The cardioid    .

                    The ray tangent to the two curves forming the cusp at the origin makes an angle  .

                    A "zoom in" of the curve   .

                    The ray tangent to the two curves forming the cusp at the origin makes an angle  .

We are really really done.   

Aside.  We can re-label the center of the circle and the angle of the tangent line as shown in Figure 11.62.

                    The circle      with center  .

                    Rays tangent to the circle at the origin make an angle    and  .

                    Remark.  The image of these two rays will be a single ray making an angle  .

                    The cardioid    .

                    The ray tangent to the two curves forming the cusp at the origin makes an angle  .

                    A "zoom in" of the curve   .

                    The ray tangent to the two curves forming the cusp at the origin makes an angle  .

Again, An Interesting Fact.   

        To obtain the standard form of a cardioid use the trigonometric identity  ,  and write

                      
                              
Now make the substitution   and get the standard form of a cardioid:

                    .  

This solution is complements of the authors.

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