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Suppose that I produce an image of a dog using a converging lens. I can draw ray diagrams for the nose of the dog as well as the back leg. These are definitely longitudinal points, not transverse. However, one typically captures a two-dimensional image of the dog using some screen. But isn’t this dog really a three-dimensional image? If so, how would one capture such an image?

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Yes, the image is 3D, though the 2D approximation we get by using a screen is pretty good.

Because the different parts of the dog are at different distances from the lens the images of those parts of the dog are also formed at different distances from the lens. So the image is a faithful 3D replica of the dog. However we can move in or out from the point of perfect focus and still get a pretty good focus. This is the origin of depth of field. In fact it can be hard to work out exactly where the focus is sharpest, especially since in many cases the difference in distance between e.g. the dog's nose and its ears is small compared to the average distance of the dog from the lens. So if we put a screen in the average position of the dog's image the whole dog will appear to be in focus even though it's frontmost and rearmost parts will actually be slightly out of focus.

The obvious way to record 3-D images is to use a hologram instead of a normal camera. Although it is in priciple possible to get the 3-D image with a normal lens, in practise it isn't possible to find the point of perfect focus precisely enough.

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    $\begingroup$ It's worth mentioning that "Light-Field" cameras are being developed. For example lytro.com/camera $\endgroup$ – Brandon Enright Dec 16 '13 at 20:47
  • $\begingroup$ @John Rennie: Great answer. I've heard of "depth of field" but didn't know what it was until your link on it. $\endgroup$ – Jesus Dec 16 '13 at 22:01
  • $\begingroup$ @BrandonEnright Another different device that also infers depth information - or, equivalently, the magnitude and phase of an image on one plane is the wavefront camera - used as a low cost, coarse measuring alternative to an interferometer. $\endgroup$ – WetSavannaAnimal Dec 16 '13 at 23:47
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Yes an ordinary lens does form a three dimensional image of a three dimensional object; but it will be a distorted image. The distortion is fundamental, and cannot be eliminated. This happens, because the "axial" or "longitudinal" magnification is equal to the square of the "transverse" or "lateral" magnification. So if a small (compared to the focal length) object, is magnified by 3x laterally, the image will be nine times longer, in the axial direction. Only when the magnification is equal to one (1) will the longitudinal magnification also be one (1). And since the three d object is not all at the same distance from the lens focal point, the magnification cannot be the same at both ends.

It is inherently impossible; no matter what, to form a REAL undistorted three dimensional image of ANY REAL three dimensional object.

There is just one unique optical system, which can form an accurate, undistorted replica three dimensional image, of a three dimensional object, and it can do that only for one specific three dimensional object; namely a portion of a spherical shell object with zero shell thickness. And the image is a virtual image of a real object; or a real image of a virtual object. The lateral magnification is equal to the square of the refractive index ratio between the two media involved. The object and image are in two different media, but both are on the same side of separating optical surface, which also happens to be a portion of a sphere.

No fixed optical system can produce sharp undistorted plane images of a plane object, at TWO or more different magnification ratios. It can be shown that if it can be done for two different magnifications, then it can be done for all magnifications. But it can't be done.

You can choose to have sharp images, or you can choose to have undistorted (geometrically) images; but you cannot have both sharp , and undistorted images. To do so requires that sin(A) = tan(A).

In normal lens design practice, small amounts of geometrical distortion of the image, are tolerated to get sharp images, which are much more important.

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