Question

I have a line going from (x0, y0) to (x1, y1) through a grid made of square tiles 2^n wide. I not only need to find the tiles the line intersects, but also the corresponding entry and exit points. All the SO questions on this I can find deal with “1×1” tiles without caring where the intersections occur within the tile.

The points won’t always be precisely on an integer, and in some cases I’ll use the natural floor and others I’ll want to round up. But letting it naturally floor in all cases for this is fine for now.

I found an example that eventually gets to a very simple case of raytracing with integers, but it doesn’t keep track of the intersection points and also isn’t adapted for anything but lines going through the center (assumed 0.5, 0.5 offset) of 1×1 tiles.

void raytrace(int x0, int y0, int x1, int y1)
{
    int dx = abs(x1 - x0);
    int dy = abs(y1 - y0);
    int x = x0;
    int y = y0;
    int n = 1 + dx + dy;
    int x_inc = (x1 > x0) ? 1 : -1;
    int y_inc = (y1 > y0) ? 1 : -1;
    int error = dx - dy;
    dx *= 2;
    dy *= 2;

    for (; n > 0; --n)
    {
        visit(x, y);

        if (error > 0)
        {
            x += x_inc;
            error -= dy;
        }
        else
        {
            y += y_inc;
            error += dx;
        }
    }
}

How can it be adapted to find the intersected 2^n x 2^n grid tiles while also grabbing the 2 relevant intersection points? It seems the ability to start “anywhere” in a tile really mucks up this algorithm, and my solutions end up using division and probably setting things up to accumulate error over each iteration. And that’s no good…

Also I think for the first and last tile, the endpoint can be assumed to be the “other” intersection point.

Answer 1

There is useful article “Fast Voxel Traversal Algorithm…” by Woo, Amanatides.
Look at the practical implementation (grid traversal section) too.
I’ve used this method with good results.