Ok, getting back on the ball here, last time I closed with a problem. If I tell you that the bottom edge of the square in the following photograph is actually 3 feet long, then can you tell me how far away the photographer (Jan Dibbets) was standing?

Ivo responded, “I do think that it’s possible to determine everything, given the height of the observer.” His argument was based on jumping immediately into the 3 dimensional space in which the camera lives.  Although this is a very sensible approach, I thought I would give another “perspective” on the problem.To begin, let’s mark off a couple of lines on the photograph:

The big horizontal line is my quick estimation for the horizon line in this photograph.  One of the key breakthroughs of the Renaissance artists was the invention of one-point perspective.  The idea behind one-point perspective is that a family of parallel lines appear to converge towards a single point on the horizon.  This device girds the composition of Raphael’s School of Athens, along with innumerable amateur drafts of hallways and streets.  Here, we can use one point perspective to observe that the two diagonal red lines above must “in reality” be parallel to one another.  By carefully placing their point of intersection along the central line of the photograph, we can further ensure that they lie perpendicular to the top and bottom of Dibbets’ “square.”  In this way we form a rectangle on the physical ground.  If we redraw our current diagram from an overhead perspective, we get the following picture:

Now notice that the two previously parallel lines of Dibbets’ “square” intersect if extended (dotted lines).  In fact, the point in which they intersect should lie underneath Dibbets’ camera.  That is, it should lie on a line perpendicular to Dibbets’ line of focus/sight.  Even though I’ve gone to a lot of trouble to avoid thinking in 3d, I still think that there’s at least one more helpful picture to have around—a cross-section of the whole setup.

There are a lot of ratios in all of these diagrams.  Although I haven’t checked too carefully, I believe that the ground distance to the square is not uniquely determined by these diagrams.  However, if we specify almost any other non-trivial measurement, then all of the free variables should be fixed.

If anyone has any more thoughts on this problem, please do share…