First attempt to fix the problem

The problem is a lack of resolution around the point c. So I just add an extra bitmap to get more resolution in that one place. It has double resolution and size 0.4 (one tenth of the side length of the main bitmap).

Sets which used to have a problem in the centre

(-1.31,0): Now with less of a problem in the centre.

New, Improved Applet

The result

It's a very slight improvement. (-1.31, 0) is greatly improved. (-2,0) and (0,1) are still pretty horrible.

Source code
Part 3...