I think there are 3 pieces to this.

calculate the intersection of the 2 lines, and hold on to the X and Y coordinates of that point find the section that the given point is in. This should be easy enough, because you have the slope of the 2 lines, and the slope of the line created by the given point and the point of intersection. Call them m_line1 , m_line2 and m_intersect . If m_intersect There's a formula to figure out the section using these values and the location of the given point. find the closest integer. There is also a straightforward calculation of this once you know the values from #1 above, and the slopes from #2. You can brute-force it, but there is an elegant mathematical solution.

These are the steps I took, at least.

Updated to add more substance

OK, I'll start us off with a discussion on #2.

If you calculate the slope of the given point and the intersection point, then you arrive at m_intersection . This is the slope of a line that passes through the intersection point. Let's assume that m_line1 is the greater of the 2 slopes, so that line1 is "above" line2 as x increases after the intersection. It makes it easier to think about for section names. We'll call section A the section given by the sliver between line1 and line2 for x larger than the intersection coordinate x, and then we'll name the other 3 sections clockwise, so that A and C are opposite each other.

If m_intersection is between m_line1 and m_lin2 , then it must be in one of the 2 sections A or C. Which section is a simple test of the x coordinate value against the intersection's x coordinate. We defined A to be the section with greater value. A similar calculation can be made if the slope is outside m_line1 or m_line2 .

This gives you the section that your point lies in. All you did was calculate 1 intersection (5 multiplications, 2 divisions and a handful of subtractions if you do it the traditional way), 3 slopes, and then a couple integer comparisons.

Edit #3 - back by (un)popular demand!

So here is how I calculated #3, the closest integer point to the intersection. It may not be the best, but it uses binary search, so it's O(log n) where n is related to the inverse of the difference of the line slopes. The closer they are together, the larger n is.

First, take the difference between the slopes of the two lines. Say it's 1/8. This means that from the point of intersection, you have to go out 8 units along the x axis before you are guaranteed that there is a whole integer on the y axis in between the two lines (it may be on one of the lines). Now, if the intersection itself is not on an integer x coordinate, then you'll need to step out further to guarantee that your starting point is on an integer x coordinate, but it is bounded. If the intersection is at x = 1.2, then in the above example, at worst you'd start at x = 41, then move down ~5 units along the y axis. Choose either the ceil or floor of the y value that you get. It's not terribly critical.

Now that you have a starting point, the closest point can be approximated by binary search. Your new line segment is between the intersection and the starting point, and your units of movement are multiples of that line segment's slope. Calculate the midpoint of the line segment and see if it lies in between the two lines. Add or subtract 1 to it if it is not a direct hit, and if either of those hits, cut the remaining distance in half and do it again. Otherwise search the further half of the segment.

If you don't have a slope difference < 1, I think the problem may be simpler (brute force the space around the intersection). But it's just a special case of the search above, where you don't need to step out that far to find a starting point.