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John Hart Roundness Formula

John Hart

linked from here

FWIW here is what I got with Berkovitz and screen geometry for a hemisphere of radius a (in the photo shoot), straddling the grid, and sticking out to the front at distance a.

a = sphere radius of shoot
A = sphere radius on screen
V = viewing distance to screen
a' = sphere radius on sensor
I = interoc distance
f = focal length of taking lens
N = near point of shoot
L = far point of shoot = N + a
B = stereo base
h = deviation of sphere near point on screen
h' = deviation of sphere near point on sensor
W = width of screen
W' = width of sensor

Formulae (with approximations if any):

h = IA/(V-A) (1) screen geometry, similar triangles
1/f = a/(La') (2) thin lens equation (L >>f)
B = h'LN/[f(L-N)] (3) Berkovitz (2LN/(L+N) >> f)

Now, fractional screen deviation = fractional sensor deviation
since we just project sensor image onto the screen:

h'/W' = h/W = IA/[W(V-A)] (1') , using (1)

Using (2) in (3) gives

B = [h'a/(a'L)]*LN/a = h'N/a' = (h'/W')*N/(a'/W')

= [IN/(V-A)]*[A/W]/[a'/W'] using (1')

But fractional sphere radius on screen = fractional screen radius on sensor, so

A/W = a'/W' and thus

B = IN/(V-A) . This reduces to your equation if A << V and L >> f.

Close, eh?

John Hart


 

Using the John Hart Formula

B = NI/(V-A)

A = sphere radius on screen

Human head radius is about 160mm.
Magnified by 10 on the cinema screen = 1600mm
Magnified by 20 = 3200mm
Magnified by 50 = 8,000mm

Let us keep the large cinema viewing distance of 20 meters.

x10

B = 2,000 * 65 / (2o,ooo - 1600) = 7mm

x20

B = 2,000 *65 / (20,000 - 3200) = 7.7mm

x50

B = 2,000*65 / (20,000 - 8,000) = 10.8mm

A one centimeter stereo base is still surprisingly small, but a bit better than the prediction from my simpler analysis of 6.5mm.

Roundness is magnified considerably on a large cinema screen, according to John Hart and I.

According to the John Hart formula, magnification (as provided by a telephoto lens) has a small contribution to roundness in the cinema, which is easily overcome by slightly increasing the stereo base.

There are other factors beside roundness which come in to choosing a stereo base. (link)

Already it is clear that a stereo camera with a fixed base (often 70-80mm) is very limiting, if your aim is to take undistorted stereo pictures from a variety of distances and view them on screens, also at variable distances.