Example 2.13. The transformation    usually maps vertical and horizontal lines onto portions of hyperbolas.  
(a)  Find the image of the vertical line  .   (b)  Find the image of the horizontal line  .  

Explore Solution 2.13.

Before we start, make x, y, u and v real variables.

The mapping    can be manipulated as follows.  

(a) Find the image of the vertical line  .  

(b) Find the image of the horizontal line  .  

Use Mathematica to make a graph of the mapping.

(c) 2006 John H. Mathews, Russell W. Howell