We do have one concern. In the previous analysis, we assumed the artist's viewpoint was below C'. The applet on the right shows that we could also assume that the artist's viewpoint was above C' - we would find a different vanishing point and viewing location and an equally valid spot from which to view the painting. The viewpoint obtained from our original assumptions is at O, while the viewpoint obtained from the new assumption is at K. The purple lines show a possible location for the square room using K as the artist's location. You can move the points C and D to change the position and size of the original painting and see how the new viewpoint changes.
When faced with a painting like ours, the two-dimensional art detective needs to examine the quantities A'-B' and C'-D'. Whichever quantity is larger corresponds to the line segment furthest away from the artist. (If they are equal, the artist's viewpoint is directly across from the midpoint of the back of the room.) Assuming A'-B' is larger, the detective has a choice of placing a particle at the point C' and moving that particle either down (corresponding to our old assumption) or up (corresponding to our new assumption). With only the information provided, this ambiguity is unavoidable. However more details about the subject could lead to a decision on exactly one viewing location.