Center of perspective

[T3D john bercovitz <bercov@xxxxxxxxxxx>]

As many of you will know, if you've followed my mathematical 
ravings, I'm very concerned about the center of perspective.  When 
you take a photo, you take it from a certain perspective and when 
you view it, you should view it from the same perspective or it 
will have depth distortion.  This goes for flat photos as well as 
stereo pairs, believe it or not.  I'll give a little background 
for those who are interested, and that will be followed by my 
question for the optically savvy on the list.
 
Below is an aerial view of four fenceposts arranged in a square.  
"O" is the center of perspective, the point from which the camera 
took the picture.  If this graphical representation looks funny, 
switch to Courier font.
 
 
                 B                   C
 
 
 
 
 
 
 
 
 
                 A                   D
 
 
 
 
 
 
                           O
 
 
 
As you can see, we have some angles here: AOB, BOC, COD
 
The photo will look like this:
 
 
             A     B           C     D
            | |   | |         | |   | |
            | |   | |         | |   | |
            | |   | |         | |   | |
            | |   | |         | |   | |
            | |   | |         | |   | |
____________|_|___|_|_________|_|___|_|__________
 
 
If you view this photo from its center of perspective, the angles 
AOB, BOC, COD, where O is now the location of the eye, will be 
recreated.  If you view from too close to the photo, angles AOB, 
BOC, COD will all be too large.  Vice versa is also true.
 
Why is this example of four fenceposts a lousy example?  Because 
the scale of the fencepost separations is very large compared to 
the size of the camera and so just knowing within a foot where the 
camera was located is good enough to reestablish the correct 
perspective when viewing the photo.
 
So why do I care?  I care because in macro photos, the camera is 
large compared to the size of the scene and so you need to know 
very exactly where the camera is located to know exactly where the 
center of perspective is.  And _that_ brings us to the real 
problem, a problem we've seen here recently, which is, "Where do 
we measure from?"
 
To me the intuitively obvious answer is, "From the primary 
principal point."  I say that because any ray that passes through 
the primary principal point (at some angle to the optic axis) 
exits the secondary principal point (at that same angle to the 
optic axis).  So what could be more correct?  Well, that's what I 
thought until I read the Good Book, "Optics in Photography" (ISBN 
0-8194-0763-1).  In Chapter 1, Verse 2, the Great Rudolph 
Kingslake sayeth: "...the entrance pupil of the lens is the center 
of perspective...".  What a terrible shock to my intuition!  So 
putting timidity (and fear of looking foolish) aside, I wrote to 
the Great Man Himself questioning his statement and he said, "The 
common base of all the entering light beams is, of course, the 
entrance pupil of the lens, the midpoint of which is therefore the 
Center of Perspective of the scene as viewed by the camera."  So 
he is taking the pinhole approach here: The limit of what hits the 
film is established by the periphery of the aperture.  Still not 
fully believing Dr. Kingslake, Professor of Optics at the 
University of Rochester and Director of Optical Design at Kodak, I 
tried the ad absurdum example he suggested in his letter, a 
telephoto lens.  In essence, I shot a scene with four small 
fenceposts close up and then calculated from the print where the 
center of perspective must have been and the center turned out not 
to be at the primary principal point which is way out in front of 
the lens, but rather at the entrance pupil, as the good Professor 
suggested.  [For human interest I should add that Professor 
Kingslake wrote a very kind letter.  You don't see that so often 
in one of his station in life, I think.]  The telephoto is a 
particularly good example because the primary principal point is 
so far out front that most of the object rays going through it 
miss the entrance pupil of the lens entirely and so never land on 
the film.
 
My question: What can I do to integrate these two intuitions?  
Anything?
 
Note 1:  One easy way to find the location of the entrance pupil 
of a lens is to use an SLR with a lens on it to find the distance 
to the entrance pupil of the lens in question by focusing on its 
aperture blades.
 
Note 2: In a symmetrical lens, the principal points are located in 
the centers of the pupils, obviating the whole question.  However, 
many or most SLR lenses are either retro or tele for necessity or 
convenience and that ain't symmetrical.
 
John B


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