A Simple Measure of Perspective Distortion

Consider two objects of identical size, S, one at the front of the scene (and, say, closer to the eye than any other object in the scene), and the other is at the back of the scene (and, say, farther from the eye than any other object in the scene). Suppose (i) the farther object is at a distance, f, from the eye and appears to be of size S_f after projection on the display; and (ii) the closer object is at a distance, c, from the eye and appears to be of size S_c after projection on the display. One measure of the amount of perspective distortion is the ratio: S_c/S_f, which will always be greater than or equal to 1. A ratio of 1 means no perspective distortion at all.

The sizes are inversely proportional to the distances, hence we have:

amountOfDistortion = S_c/S_f = (S/c) / (S/f) = f / c

Define the depth of the scene, Δ, as Δ = f - c, then:

amountOfDistortion = (c + Δ)/c = 1 + Δ/c

If Δ << c, there will be very little perspective distortion, and the scene will look more like an orthogonal projection than a perspective one. If Δ >> c, then there will be very exaggerated distortion (imagine looking at a tree in the distance with a pencil right in front of your eye), and the scene could appear cartoonish.

Generally we want the relationship between Δ and c to be somewhere between these two extremes. Since Δ is out of our control (it is determined solely by how the scene is constructed), our only control over the amount of perspective distortion is c: the distance we place the eye from the front of the scene.

To summarize, note the following:

Neither the field of view (ecXmin, ecXmax, ecYmin, ecYmax) nor the values for ecZmin, ecZmax, nor even ecZpp (a common misconception) affect the amount of perspective distortion we see. Our only control is how close we place the eye to the front of the scene, which we summarize next.
Since we have no control over Δ, we often choose to compute c as: c ← f * Δ for some factor, f, a reasonable value for which might be something between 1 and 2. Assuming we compute the center of attention as the middle of our scene, this would give us a distance between the eye and the center of attention as (c + Δ/2).