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I'm not too sure of what John writes below. I have written a program (part
of the E&M Animated Chalkboard package) to map electric field lines in 2
dimensions. The way it works is that I place a test charge near one of the
charges that makes up the charge distribution. I then calculate the force
on the test charge due to all charges in the distribution, the acceleration,
the velocity, and the new position taking a very small time interval. I
move the test charge to that position plotting the path to get there and
then repeat until the charge either gets far off the visible screen or ends
up at a second charge. This method produces the field line patterns we all
know.