Perspective: focal length v. subject distance or "zooming with your feet"

We all know that zooming with your lens and zooming with your feet do not produce the same results in general. However, there are important exceptions.

Consider the two photos below. The first was taken by zooming with the lens and the second by zooming with my feet. It is obvious that the two images are more or less identical after allowing for (i) small errors in alignment (both shots were handheld), and (ii) slight barrel distortion in the second image, caused by the lens. Both are the full size, uncropped images. The only processing was for brightness and colour balance.

The subject is an information poster attached to a fence around the restoration work on the lock. The first image was taken with a 100mm lens at some distance from the poster. The second image was taken with a 12mm lens at a distance that was approximately 0.12 times the subject distance for the first image.

This is quite a good illustration of the basic principles of perspective. Going closer to your subject increases the image size (which is inversely proportional to the distance). But the image size is also proportional to the focal length of the lens. So, the image size may be kept constant if the subject distance and focal length are changed simultaneously by the same factor.

This works if the subject is 2-D and in a plane parallel to the sensor in the camera. For a 3-D scene where different parts of the scene are at different distances from the camera, the second image will look the same as the first only if everything in the scene can be moved to 0.12 times its original distance from the camera. In other words, the whole scene would have to be compressed by the factor 0.12.

Uh-oh! When your subject is entirely 2D and fills the whole frame a distortion free lens is the way to go, e g. a macro or repro lens.

Well, on the bright side, you've shown that there can be extremely little 3D perspective in an 2D object - when it's nicely placed perpendicular to the optical axis of your camera and lens! 🥸

This thread makes no sense. For a flat 2D subject, it is pretty obvious that there are no pictorial perspective effects. Just, like when I used to print with an enlarger, I just moved the head up or down, or could change lens FL to fit the negative image onto the variable paper size.

What may be obvious to you and me is, unfortunately, not so obvious to many others. Most modern photographers haven't thought much about the perspective effects in using a camera to copy documents or about using an enlarger to make prints.

"Zooming with your feet" is about the dumbest expression imaginable!

I think you needed to be a bit clearer with your original post, as I did not understand the argument.

So you take two images where there is no perspective present (only magnification of a 2D plane), formulate a relationship between subject distance and image size (on sensor) and extrapolate that to explain perspective effects in the 3D world?

There is a lot of assumption in there.

There is no case where you can move the camera and record the same scene, it never happens. Perspective is the unique distortion of shape caused by viewing from a singular point. There are no two different points that give the same view. The 0.12 times hypothesis is thus impossible and therefore must also be incorrect.

I wouldn't call a 2D image parallel to the sensor plane an important exception, in fact it is explained perfectly by the same geometry of image formation. The principal is simple enough, take a portrait of your loved one from 2m away with the backdrop of a snowy mountain range say 5 miles (8km) distant. Move back to 10m for a whole body shot, because you love them completely and entirely. Now your distance from your loved one has increased by a factor of 5 and so their relative size has changed quite noticeably. The mountains in the distance however are now 8008m away and as your relative distance is roughly the same as it was before they appear relatively the same size. Use a zoom lens and it is essentially just a magnification as your position (or distance to either subject) has not changed.

But we have learnt that Gloucester Lock is closed for repairs. :-)

There are special cases in which it could happen.

Example: The first scene with the 100mm lens is in a warehouse with the camera 50 feet from a painted mural on a wall and oriented perpendicular to it. 25 feet from the camera is small flat sign (let's say one foot square) with almost no thickness, suspended by wires from the ceiling, which is out of the frame. It's oriented parallel to the wall and positioned in the exact center of the frame. Those two objects comprise everything that is seen by the camera. To shoot the same scene with the 12mm lens, move the camera to 6 feet from the wall and move the sign to 3 feet from the wall. That should work, correcting for barrel/pincushion distortion and such things. You'll want to avoid including 3D objects because the 12mm lens will render them with a higher degree of volume deformation, so the result won't be the same.

So it only works withe 2D objects parallel to the sensor plane (no 3rd dimension so no distortion in perspective as long as you keep the two parallel), and it only works if you change the position of the objects (change the scene) when you move the camera?

I don't understand, what have you proved?

It's pretty clear to me.

In his op he said

In his op he gives an example of an exception and at the end states the circumstances you would need for zooming with the lens or the feet to produce the same image for a 3D scene.

Not to me.

Yep, he said that.

But he also said:

Taking a picture of a 2D plane is an important exception in understanding the effects of perspective when walking or zooming? Let's walk through this, and also zoom through it...

Take your camera with it's fixed lens and zoom with your feet taking shots at every pace, now fix your zoom lens and from the same spot take images at increments of "zoom".

Are they all the same, or are only the two where the sign filled the sensor the same? Even in the example there is only one point where the condition is "true" and it only happens when you remove 3D perspective completely and take photos of a 2D surface that fills the frame. And from that we can extract a simple mathematical formula to describe the way perspective never works, by moving mountains and trees around the landscape to maintain the 0.12x? Then sybersitizen's example really only proves that if you have to contrive and control an example to that extent to prove a point then the point is rather contrived.

Why not describe the difference between zooming and walking by what actually happens? Seems more useful to me than equating it to something that doesn't.

Only that special cases could be constructed in which the result could be exactly duplicated by rearranging object distances as suggested in the OP.

Ah, but the first word in the thread title is "perspective" and so far we've restricted the explanations to 2D objects parallel with sensor plane so they lack any perspective.

Seems pretty obvious.

The first sentence in the body of the post is this:

'We all know that zooming with your lens and zooming with your feet do not produce the same results in general. However, there are important exceptions.'

So, we can identify the exceptions. AFAIK, all exceptions, important or not, apply to 2D subjects.

Perspective: the art of representing three-dimensional objects on a two-dimensional surface

Photographing a 2D object is basically a facsimile, there is no perspective just a copy. The op is just an exercise in distance vs focal length, basically use a longer focal length = move further away.

I still do not see the point.

What is and isn't important will depend on each individual's interest in a topic.

In a pickle barrel discussion¹, which is what I call things like this, there is often no particular point.

¹ An old expression with reference to times past when men would gather around the pickle barrel in a small-town store to shoot the breeze.