# Can Math Model Real-World Camera Focusing Dynamics?

• EmilioL
In summary, the conversation discusses the problem of modeling camera focusing movement, particularly in relation to the use of a simple lens, collimated rays, and monochromatic light. It is noted that while most camera lenses are not simple, this is a common approximation. The conversation then delves into the classical formula for achieving the sharpest focus, which involves a relationship between the back distance, object distance, and focal length. However, in real cameras, the focus is achieved by moving the lens instead of the back, leading to a more complex equation. The conversation ends with a request for a more efficient algorithm for finding the exact solution for delta, the focusing distance added.
EmilioL
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!

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.

## 1. How does camera focusing motion work?

Camera focusing motion involves adjusting the lens of a camera to bring the subject into sharp focus. This is achieved by changing the distance between the lens and the image sensor, which allows the camera to capture a clear and detailed image.

## 2. What is the math behind camera focusing motion?

The math behind camera focusing motion involves the use of geometric optics, specifically the thin lens equation. This equation takes into account the focal length of the lens, the distance between the lens and the subject, and the distance between the lens and the image sensor to determine the necessary adjustments for achieving focus.

## 3. Can you explain the concept of depth of field in relation to camera focusing motion?

Depth of field refers to the range of distance in an image that appears acceptably sharp. In camera focusing motion, adjusting the focus changes the depth of field, with a smaller aperture resulting in a larger depth of field and a larger aperture resulting in a smaller depth of field.

## 4. How does autofocus technology work in camera focusing motion?

Autofocus technology uses a combination of sensors, motors, and algorithms to automatically adjust the focus of a camera. These sensors detect the contrast and sharpness of the subject, and the motors then move the lens accordingly to achieve focus. The algorithms help to determine the best focus point and make adjustments as needed.

## 5. Are there any tips for improving camera focusing motion?

Some tips for improving camera focusing motion include using a tripod for stability, using a single focus point for more control, and using manual focus for precision. It is also important to understand the limitations of your camera's autofocus system and make adjustments accordingly, such as using manual focus in low light situations.

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