[MF3D.FORUM:95] Image Size

[Richard Rylander <rlrylander@xxxxxxx>]

Not content to leave well enough alone...

The request from John B to find the solid angle subtended by various
image formats has been simmering on my back burner for a while, so I
hauled out the old analytic geometry text and took a stab at it.  Turns
out, the final equation I came up with is fairly simple.

     [ All angles will be in radians, solid angles in steradians. ]

Consider a rectangle "w" units wide, "h" units high, viewed from a point
"f" ('focal length' of viewer lens) units away from the center of the
rectangle.  The "1D" angular width, "aw", and angular height, "ah", [the
easy calculations] are:

    aw = 2*arctan(w/(2f))         ah = 2*arctan(h/(2f))

The "2D" solid angle, "s" is a little trickier:

    s = 2*pi - 4*arccos(sin(aw/2)*sin(ah/2))

In the limit as "f" approaches 0 (your eye virtually in contact with the
slide), the solid angle "s" approaches 2*pi steradians : a hemisphere.

In another message, John B. felt that rectangular formats should not
score as high as square formats of the same solid angle measure.  I
initially suggested that the solid angle be divided by the aspect ratio
of the format (always expressed as a number >= 1), but it turns out this
penalizes rectangles too much - place two identical squares side-by-side
and the solid angle is not quite double that of a single square (a
"cosine effect" due to the increasing obliquity of the format edges)
while the aspect ratio is now 2, so the "adjusted" solid angle of the
bigger rectangle is less than that of the original square (not fair, not
fair at all).

A more reasonable adjustment is to divide the calculated solid angle by
the square root of the aspect ratio (it might be argued that this
reflects the 2D nature of format areas relative to linear dimensions of
the format).  Solid angles change in funny ways particularly as you
approach the 2*pi hemisphere limit, but for typical fields of view this
seems to work fairly well (formats that are nearly square are affected
very little).

A final step is to scale the result to a number people have a better
feel for (steradians are pretty alien to most).  I would suggest
defining an "Immersion Factor" as the [adjusted] solid angle divided by
2*pi to produce a measure that is effectively the fraction of a
hemisphere that a format subtends.

The table below takes the format sizes, viewer focal lengths used by
Alan Lewis and calculates the solid angle, adjusted solid angle, and
"Immersion Factors".

Format      Size      Focal Length  Solid Angle     Adj. Solid Angle
Immersion Factor
5P          23 x 21          44             2.0896
1.9967                 0.3178
7P          23 x 28          44             2.4174
2.1910                 0.3487
MF        50 x 50          78             2.6838
2.6838                 0.4271
3Disc     24 x 36          43             2.7753
2.2660                 0.3606
MF-W   56 x 58          78             2.9964
2.9443                 0.4686
6 x 7      60 x 70          78             3.2876
3.0437                 0.4844

Does "Immersion Factor" = "WOW" ?  This seems to support the various
MF's.

Time to break out those programmable calculators.

Richard Rylander