Maximum Projection FIeld of View

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What is the maximum field of view that can be accomplished via a projection matrix with no distortion? There is a hard limit of < 180 degrees before the math completely breaks down, but experimenting with 170-180 degrees leads me to believe that distortion and deviation from reality begins prior to the hard limit. Where does the point at which the projection matrix begins to distort the view lie?

EDIT: Maybe some clarification is in order. As I increased the FOV angle toward 180 with a fixed render size, I observed objects getting smaller much faster than they should in reality. With a fixed render size and the scene/camera being identical, the diameter of objects should be inversely proportionate to the field of view size, if I'm not mistaken. Yet I observed them shrinking exponentially, down to 0 size at 180 degrees. This is undoubtedly due to the fact that X and Y scaling in a projection matrix are proportionate to cot(FOV / 2). What I'm wondering is when exactly this distortion effect begins.

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math3dperspectivecamera
Okay, I think I have an idea where my logic went wrong. I was reasoning in terms of absolute angles, as if projecting onto a sphere around the view. An object twice as large casts twice the angle from the viewing point, and doubling the field of view halves the relative angle of the object. But in 3D graphics this is being projected onto a plane that is not perpendicular to the rays from view to object like a sphere is, so the relative angular size isn't necessarily proportionate to size on the plane. – Justin Olbrantz

1 Answers

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Short answer: There is no deviation from reality and there is always distortion.

Long answer: Common perspective projection matrices project a 3D scene onto a 2D plane with respect to a camera position. If you consider a fixed distance of the plane from the camera, then the field of view defines the plane's size. Larger angles define larger planes. If you fix the size, then the field of view defines the distance. Larger angles define a smaller distance.

Viewed from the camera, the image does not change whether it sees the original scene or the plane with the projected scene (i.e. there is no deviation from reality).

Problems occur when you look at the plane from a different view point. E.g. when the projected plane is displayed on the screen (fixed size), there is only one position of the camera (your eye) from which the image is realistic. For very large field of view angles, you'll need to be very close to the screen to find that position. All other positions will not result in the correct image. For small field of view angles, the resulting distortion is very small and users will mostly consider it a realistic projection. That's because for small angles, the projected image does not change significantly if you change the distance slightly (changing the distance from 1 meter to 1.1 meters (10%) with a small fov is less problematic than changing the distance from 0.1 meters to 0.2 meters (100%) with a large fov). The most extreme case is an orthographic projection with virtually zero fov. Then, the projection does not depend on the distance at all.

And there is always distortion if objects are not at the projection axis (i.e. for any fov greater than zero). This results in spheres not projecting to perfect circles. This effect also happens with small fovs but there it is less obvious.