Can Math Model Real-World Camera Focusing Dynamics?

AI Thread Summary
The discussion centers on modeling camera focusing dynamics, particularly the challenges posed by real-world lens mechanics compared to theoretical models. It highlights that while the classic lens formula applies at photographic infinity, it becomes complex for close-up photography due to the movement of the lens rather than the film or sensor. The equation 1/B + 1/D = 1/f is modified to account for the additional distance (delta) when focusing on nearby objects, but finding an efficient solution for delta remains a challenge. The conversation emphasizes the lack of resources in photography literature addressing this specific problem, as many companies treat their focusing systems as trade secrets. Overall, the thread seeks guidance on deriving a direct solution or a more efficient algorithm for calculating the necessary adjustments in lens positioning.
EmilioL
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This problem arose in modeling camera focusing movement, such as a control system might do.

It assumes a simple (thin) lens, rays close to the optical axis, and monochromatic light. While most camera lenses are not simple, this is a first approximation.

Camera lenses project an image of a distant object (the subject of the photo) on a screen (the film or digital sensor). When the object is very far away (at "photographic infinity") the rays coming from it are nearly parallel (collimated), then the back distance (from lens to film/sensor) tht gives the sharpest image is equal to the focal length of the lens (by the definition of focal length). But when the object is nearby, the back distance must be increased to bring the projected image into focus.

Clasically, the image will be in sharpest focus when the relationship between the back distance, object distance and focal length is 1/B + 1/D = 1/f. However, real cameras do not focus by moving the back (film or sensor), they focus by moving the lens forward, towards the object. This increases B, but also decreases D by the amount. When D is large, this decrease is insignificant and can be (and usually is) ignored. But in close-up photography, and especially extreme close-up (macro lens) photography, the difference can be significant.

Starting from the lens in its infinity focus position, and calling the focusing distance added (i.e., additional bellows extension) delta, the above formula becomes 1/(f + delta) + 1/(D - delta) = 1/f.

It's easy to devise an algorithm that gives an approximate solution: loop, incrementing delta by a fixed amount until the equation becomes true.
But that's inefficient and doesn't give an exact soltuion. Increasing the precision of the algorithm by using a smaller increment also increases run time..

Given f and D, is there a direct solution for delta? Failing that, is there a more efficient algorithm?

I'm sure this is obvious to somebody here, but not to me. Any help would be greatly appreciated. I realize this is a math problem--the connection with physics is that the Gaussian focus equation -- given in every physics textbook and ever optics textbook ever written -- turns out to be somewhat difficult to apply to real cameras, which focus by moving the lens, not the back. I checked dozens of physics and photography books,, and none that I found discuss this problem. Photography doesn't have refereed journals and photography companies consider their control systems to be trade secrets. Thanks in advance!
 
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Immediately after posting the above, I realized that distance could be measured to the film/sensor plane--
which does not move. The focus equation 1/B + 1/D = 1/f becomes 1/(f + delta) + 1/(D - (f + delta)) = 1/f
Unfortunately, this isn't any easier to solve, so far as I can see.
 
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