I suggest that you break the query into two parts, points_inside(parallelogram) := points_inside(triangle1) union points_inside(triangle2), where the triangles are formed by T(origin,origin+A,origin+A+B) andT (origin,origin+A+B,origin+B).

Regarding the find-all-points-in-a-triangle query, this is basically a scanline-conversion problem. You can find solutions in computer-graphics like sources (for drawing triangles onto grids of pixels, that sort of thing).

One of the (many) hits for "scanline conversion of a triangle":

http://vis.uky.edu/~ryang/Teaching/CS335-spr05/Lectures/g_05_fill.pdf

You can probably find better ones. You could probably work with the polygon directly, it'll just be a little more complicated and might be a bit harder to find good tutorials.

I suggest that you break the query into two parts, points_inside(parallelogram) := points_inside(triangle1) union points_inside(triangle2), where the triangles are formed by T(origin,origin+A,origin+A+B) andT (origin,origin+A+B,origin+B).

Regarding the find-all-points-in-a-triangle query, this is basically a scanline-conversion problem. You can find solutions in computer-graphics like sources (for drawing triangles onto grids of pixels, that sort of thing).

One of the (many) hits for "scanline conversion of a triangle":

http://vis.uky.edu/~ryang/Teaching/CS335-spr05/Lectures/g_05_fill.pdf

You can probably find better ones. You could probably work with the polygon directly, it'll just be a little more complicated and might be a bit harder to find good tutorials.

You could also try drawing the edges into a small image, then do some floodfill-like search until you hit them. Bresenhams is the prototypical algorithm for scanline conversion of a line.