Viewing Angles and Image Areas

[Robert Camfield]

At a common focal plane-subject distance, the indexes below reflect the differences in image area covered by various lens focal lengths, with the index for the standard focal length (50mm) set to unity (1.000). Note a couple of things. For focal lengths longer than 50mm, the image area declines at twice the change in viewing angle (VA). For longer lenses, this well known relationship is linear, in close approximation. The VA of the 105mm focal length (23 degrees) is approximately one half that of the standard, while the image area for the 105mm declines somewhat greater than four times (4x), to 0.230, compared to 50mm (VA=46 degrees). Similarly for longer lenses: the index for 200mm (VA=12 degrees) is 0.061. For focal lengths shorter than 50mm, the relationship is highly non-linear. The VA for the 20mm lens (94 degrees) is approximately twice that of the 50mm, whereas the image area of the 20mm increases 6.382x compared to the 50mm.

Example Focal Lengths

Focal Length     Viewing Angle     Index of              Elasticity                                                 Image Area 600mm               4.0               0.007               2.00 400mm              6.0              0.015               2.00 180mm             13.5              0.078               2.03 105mm             23.0              0.230               2.07 50mm              46.0              1.000               2.33 28mm             75.0              3.268               2.82 20mm             95.0              6.382               3.41 15mm           110.0            11.320               4.36 The trigonometric calculations -- essentially, tangents to angles of two adjacent right triangles -- are straightforward. Thought you might have interest in this. Also, my apologies for the bad formatting of the table - …Robert

[Erik Lund]

Thank you.
Very useful to remember!

[Robert Camfield]

Erik,

I just noticed...the table contains an error: the viewing angle for 400mm is 6 degrees, not 12 degrees as stated.

Robert
I have corrected the error in the original post - Erik

[Roland Vink]

What does "Elasticity" mean?

[Robert Camfield]

Roland,

Elasticity is the ratio of the change in image area to the change in viewing angle, measured in logs with respect to the adjacent shorter focal length. As an example, the elasticity for the 35mm focal length is calculated as ln(35mm imagine area/28mm image area)/ln(VA of 35mm/VA of 28mm). Elasticity can be interpreted as a measure of sensitivity of image area to viewing angle. 

Elasticity for a defined focal length can also be calculated as a near mid-point of the arc over the adjacent longer and shorter focal lengths. In the case of 35mm, this approach would determine elasticity as the (approximate) mid-point of the arc for 50mm (adjacent longer lens) and 28mm (adjacent shorter lens). The difficulty with this approach is that the adjacent "long side" focal length may be much closer to (or further from) the focal length of interest than the adjacent "short side" focal length...at least for well established focal lengths: 18, 20, 24, 28, 35, etc. To deal with this estimation error, I have simulated viewing angles for very near constant-distance focal lengths for the established focal lengths, and then estimated the image areas and corresponding elasticities - which can surely improve the accuracy of the estimated elasticities. For example, for 35mm, I would use simulated viewing angles for 33mm and 37mm, and then calculate the arc image areas and the commensurate elasticities.

Robert