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The question in case of TL;DR: Given an extremely long distance (or focal length), would the image generated of a given building or buildings (or indeed of any real world object) always specifically resemble orthographic projection because of some way that our eyes/cameras work, or is it possible for it to resemble an oblique projection as well?


Some context:

We know that because of perspective projection, parallel lines appear to converge to a point. However, the further away we are, they start to appear less convergent and more parallel.

You can see examples of this very “parallel-like projection” in buildings which are very far away (especially if you use a telescope), and shadows which are cast by the sun, a rather distant star from the earth. You can also see this effect happen when you adjust the focal length of your camera.

This animation should give you a good idea of what I'm talking about. It's from the Wikipedia article on focal length:

https://commons.wikimedia.org/wiki/File:Camera_focal_length_distance_house_animation.gif

And the accompanying caption:

In this computer simulation, adjusting the field of view (by changing the focal length) while keeping the subject in frame (by changing accordingly the position of the camera) results in vastly differing images. At focal lengths approaching infinity (0 degrees of field of view), the light rays are nearly parallel to each other, resulting in the subject looking "flattened". At small focal lengths (bigger field of view), the subject appears "foreshortened".

EDIT: When the centre of projection is an infinite distance from the projection plane, the projection lines are truly parallel and it is termed a parallel projection. The kind of parallel projection it is depends on what angle the projection lines interesect the plane at. If they intersect the plane at a perpendicular angle it is an orthographic projection. If they intersect the plane at an angle that is not 90° then it is known as an oblique projection.

Here's another picture of what these projections typically look like, with the addition of perspective projection:

enter image description here


Imagine you are in a plane up in the sky, very high above the surface of the earth. Or perhaps you are at the top of the Burj Khalifa. Or maybe, you can take pictures from a satellite orbiting the earth. The point is that you are an observer very far away, so the parallel lines don't converge very much.

Anyway, you look down and see the buildings below you.

To restate the question: Given an extremely long distance (or focal length), would the view of these buildings (or indeed any real world object) always specifically resemble orthographic projection because of some way that our eyes/cameras work, or is it possible for it to resemble an oblique projection as well?

There are plenty of examples of oblique projection when you look at shadows being cast onto a surface. Because the sun's rays are approximately parallel, when it is not directly overhead shadows it casts of objects are oblique projections. However, I haven't found any examples in the context of optics.

Basically, what I'm asking is would the scene below you ever look something like this:

enter image description here

enter image description here

A closeup of the same painting. Notice the angles of the building lines; it looks most similar to the oblique cavalier projection in the previous diagram of the 3D houses.

I have seen examples of things in oblique projection in technical drafting and in computer graphics. The same is true of orthographic projection. Not only that, with a telecentric lens you can take pictures of objects which will be in orthographic projection. And as I mentioned before, when you move very far away a similar effect happens.

What I have noticed however, the resulting image tends to be an orthographic projection, and an oblique projection doesn't even seem possible. Additionally, I haven't found any pictures of oblique projection which have been taken with a telecentric lens, only pictures like this:

enter image description here

Which is definitely orthographic (you can try tracing lines over it in an image editor).

Thank you for reading this far and I hope to hear from you.

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    $\begingroup$ I don't know what the terms "oblique projection" and "orthographic projection" mean. $\endgroup$ Jun 20, 2022 at 2:39
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    $\begingroup$ @GerryMyerson Orthographic projection = projection lines intersect plane at perpendicular angle. Oblique projection = projection lines intersect plane at an angle which is not 90°. Each one produces rather different images. I'll add this to the main question as well. $\endgroup$ Jun 20, 2022 at 3:02

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The picture of an object through a typical extreme telephoto lens is orthographic (or at least nearly so) because the lens projects the image onto the center of the frame of the film(*) at an angle perpendicular to the plane of the film and the lens is so far from the film that the angle at which it projects other parts of the image is nearly perpendicular to the plane of the film even at the edges of the frame.

Since all the projection lines are (nearly) perpendicular to the film, you get something that is (nearly) an orthographic projection. In principle there will still be vanishing points for some sets of parallel lines, but in practice this will be difficult to detect.

An alternative technique that is possible with a view camera, with a shift lens, or with a tilt-shift lens is that you can shift the lens up, down, or sideways relative to the center of the frame of the film. A technique called perspective control uses this capability to make lines in the image parallel that otherwise would not be parallel.

Perspective control seems usually to be applied mainly to the vertical lines of buildings, but if you were to take a photograph of a building from high above the building, far to the side of the front-to-back axis of the building, you might be able to shift the lens far enough downward and to the side so that the plane of the film is parallel to one face of the building. Having shifted the lens like this, the lens will project part of the building onto the center of the film's frame at an oblique angle.

If you are also far enough away from the building that the angle between light rays reaching the lens from the building is very small, then all the visible points of the building will be projected at (nearly) the same oblique angle onto the film, producing a (nearly) oblique projection.

As a workaround, if you do not have a camera or lens system capable of shifting the lens with respect to the film, you could put the subject in a lower corner of the frame of an ordinary camera and crop to the subject so that the part of the frame that you use in your photograph is a part that received its image all at nearly the same oblique angle. I think this is better to do with a lens that is not too far from the film since you want the projection in that corner of the film to be far from perpendicular to the plane of the film.

Here is an example of this workaround:

enter image description here

The extreme cropping that is necessary for this workaround also means that you lose resolution, which is a large contributor to the poor quality of the image. But I think it is clear enough to see that the front of the rectangular block is a perfect rectangle in the image, while the lines of the block perpendicular to the front are projected onto (nearly) parallel lines at a diagonal angle, just as they are in the oblique-projection drawings of buildings in the question.

(*) I write "film" here for the object that takes the photographic image. In cameras before the digital age, this was either photographic film or a photographic plate, but in a digital camera, this object is an electronic sensor. The geometry is the same in each case, however, and I found the word "film" convenient to use.

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  • $\begingroup$ Thank you for your detailed response. I suppose what I was wondering was how people centuries ago got the inspiration to depict things in oblique projection given it's not the easiest projection to obtain. I wonder how feasible it would be to do what you have described with a camera obscura as people have known about that since antiquity. Perhaps some practical application of descriptive geometry will help me to find out. $\endgroup$ Jun 21, 2022 at 9:07

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