Finding an image under a given transformation

[Miike012]

My question is in the paint document.
And I think I know the answer to my question. I asked why can't I let v = 1 then my first first region transformation would

the line y = b between -a≤x≤a.

The reason I think I can't do this is because the end point v = 1 is a point and not a line segment...

 

[LCKurtz]

The graph is bounded by the rectangle $|u|,|v|\le 1$ but it isn't that rectangle. It is just contained in it. And the other graph is contained in the image of that rectangle, which is $|x|\le a,|y|\le b$.