Simplified Bresenham's line algorithm: What does it *exactly* do?

Question

Based on Wikipedia\'s article on Bresenham\'s line algorithm I\'ve implemented the simplified version described there, my Java implementation looks like this:

Answer 1

There are various forms of equations for a line, one of the most familiar being y=m*x+b. Now if m=dy/dx and c = dx*b, then dx*y = dy*x + c. Writing f(x) = dy*x - dx*y + c, we have f(x,y) = 0 iff (x,y) is a point on given line.

If you advance x one unit, f(x,y) changes by dy; if you advance y one unit, f(x,y) changes by dx. In your code, err represents the current value of the linear functional f(x,y), and the statement sequences

    err = err - dy;
    x1 = x1 + sx;

and

    err = err + dx;
    y1 = y1 + sy;

represent advancing x or y one unit (in sx or sy direction), with consequent effect on the function value. As noted before, f(x,y) is zero for points on the line; it is positive for points on one side of the line, and negative for those on the other. The if tests determine whether advancing x will stay closer to the desired line than advancing y, or vice versa, or both.

The initialization err = dx - dy; is designed to minimize offset error; if you blow up your plotting scale, you'll see that your computed line may not be centered on the desired line with different initializations.

Answer 2

Just want to add one bit of "why" information to jwpat's excellent answer.

The point of using the f(x) = dy*x - dx*y + c formulation is to speed up the calculation. This formulation uses integer arithmetic (faster), whereas the traditional y = mx + b formulation uses floating point arithmetic (slower).