The idea. The world is a grid of unit cells, each empty or a wall. For screen column i of N the ray direction is dir + plane * c with c = 2(i + 0.5)/N - 1 and |plane| = tan(FOV/2). The wall strip height in pixels is h = f / d where f = (W/2) / tan(FOV/2) is the focal length in pixels and d the wall distance.
DDA grid traversal (Amanatides and Woo). Instead of sampling along the ray, jump from grid line to grid line. With a unit ray (dx, dy) the ray moves 1/|dx| to cross one column of cells and 1/|dy| to cross one row. Keep the distances to the next vertical and horizontal grid lines, always advance the smaller one, and stop at the first wall cell. It visits each crossed cell exactly once, never skips a corner, and gives the exact distance. The number of steps is at most |x1 - x0| + |y1 - y0| cells.
Fisheye correction. The DDA distance is measured along the ray (Euclidean). A flat wall straight ahead would look curved if its height were taken from that length, because edge rays are longer by 1/cos(theta). The correct distance for projection is the perpendicular one, d = t cos(theta); with theta = atan(tan(FOV/2) c) this is exactly t / |dir + plane c|. Switch the fix off with a wide FOV to see the bulge.
Texture coordinates. The hit point on the wall gives u = frac(x_hit) (or y_hit), mirrored on two of the four wall faces so the texture reads correctly; the row inside the strip gives v. Floor and ceiling are cast row by row: a row at y pixels below the horizon lies at perpendicular distance 0.5 f / y and its world position is stepped linearly across the row. Walls on the y-facing sides are darkened by 30 percent and distance fog is 1/(1 + 0.14 t).
Checks. Check 1 compares the DDA with the closed-form distance to the walls of an empty rectangular room, using different code (a division formula), on 20 000 random rays. Check 2 shows the naive marcher always overshoots by between 0 and one step, so its mean error is about step/2. Check 3 gives the theoretical ratio of edge to centre ray length, 1/cos(FOV/2), and measures it with the real DDA.
Honest simplifications. Walls have constant height and sit on a grid; floors and ceilings are flat; there are no sprites, doors, slopes, looking up and down, reflections or shadows, so this is a teaching model of the Wolfenstein-style technique, not a general renderer. Texture sampling is nearest-neighbour with no filtering, so walls alias at distance on purpose. Performance numbers depend on your device.