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It is more on theoretical-math-physics stuff side, it is not crucial for practical application (the practical context of this question is "video projection"), just curiosity and liked to hear somebody's take on what I see as inconsistency within photometry related definitions.

This year I have been faced twice with foot-lamberts (which I do not normally use, SI units here :)), due to some "imperial" projects and although practically they are over and there were no issues, they left me with a question I can not get off my mind.

Now, there are two definitions of foot-lambert that I can find, simplified:
DEF1) a unit of luminance equal to one lumen per square foot [4][5][6]
DEF2) a unit of luminance equal to 1/π candela per square foot [1][2][3]

I see DEF1 been used all around. Practically it is on every random online article, manufacturer sites, and it was what US based partners used - "just take projector lumens and divide it with projection surface area in feet squared".
I see DEF2 been used in Wikipedia, AVIXA CTS definition and any source that seemingly has some peer reviewing / scientific credibility.

Now, if I examine he differences between the definitions ->

DEF1 foot-lambert calculation is basically projector lumens over projection surface measured in ft2.
That gives DEF1 ftL units of [lm/ft²].
Luminous flux, measured in lumens [lm], ties to luminous intensity, measured in candela [cd]. One candela equals one lumen per steradian, or [cd] = [lm/sr]. And one lumen equals one candela steradians, or [lm] = [cd⋅sr].
Thus DEF1 ftL units can be expressed both as [cd⋅sr/ft²] and [lm/ft²].
It refers to illuminance in its definition.

DEF2 says "the footlambert (fl) is a U.S. customary unit of measurement for luminance. It is equal to 1/π candela per square foot." and "A foot-lambert equals 1/π or 0.3183 candela per square foot, or 3.426 candela per square meter (the corresponding SI unit)."
Thus DEF2 proposed ftL unit is [cd/ft²] (applying constant [cd/ft²]/C, where C is 1/π, or [cd/m²]/C, where C is 3.426 does not change units).
It refers to luminance in its definition.

Obviously units between DEF1 and DEF2 differ.
DEF1 [cd⋅sr/ft²] is not the same as DEF2 [cd/ft²], as solid angle in DEF2 is thrown away (or constant 1/π sr is assumed?).
DEF1 by its units seems to measure illuminance. For me it seems the same as lux, only here it is [lm/ft2] instead of [lm/m2].
DEF2 by its units seems to measure luminance. For me it seems the same as nits, only here it is [cd/ft2] instead of [cd/m2].

What's going on here, what am I missing?
Like literally everybody (a safe overgeneralisation in the field of video solutions :)) using DEF1.
Why such DEF2 exists? Maybe it is measuring illuminance due to that "1/π" note, implicitly assuming 1/π solid angle due to historical/technical aspects, is that it?

[1] Wikipedia https://en.wikipedia.org/wiki/Foot-lambert
[2] Definition in AVIXA CTS exam materials.
[3] Efunda https://www.efunda.com/glossary/units/units--luminous_luminance--foot-lambert.cfm
[4] Dictionary https://www.dictionary.com/browse/foot-lambert
[5] Dictionary https://www.thefreedictionary.com/foot-lambert
[6] Placeholder for google result on "projector calculating foot lamberts". :)

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  • $\begingroup$ The units of foot-lambert are really [lm * sr / ft^2]. Your "DEF1" comes from links 4, 5, and 6. The langauge in link 4 is super confusing to read. It says to take the luminance of "a surface emitting a luminous flux of one lumen per square foot", which is then redefined as "the luminance of a perfectly reflecting surface receiving an illumination of one foot-candle". This def refers to luminance and illuminance. To me it's the same as DEF2, just worded terribly and avoiding a discussion of units. Link 5 actually contains both defs! $\endgroup$ Oct 28 '20 at 0:54
  • $\begingroup$ In other words, DEF1 is not wrong per se, just really confusing. "The luminance of " some other derivation of illuminance/intensity is easy to get tripped up by. I find DEF2 cleaner, myself. $\endgroup$ Oct 28 '20 at 0:58
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I would strongly recommend staying with the NIST SP811 definition of a foot-lambert:

candela per square meter (cd / m2) ....... 3.426 259 E+00

You can find this document here, among other sites.

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  • $\begingroup$ Thanks for the link! NIST defines lambert as being linearly tied to candela per square meter in Appendix B.8 and Appendix B.9. foot-lambert is not defined there. By looking other definitions and how adding foot "works" there deducing foot-lambert being "candela per square foot". That would be similar to DEF2, although the "1/π" would be missing. => people working with video projection technology in U.S. have their own-inside-alternative "foot-lambert" definition as "lumens per square foot" is something totally different than "candela per square foot" physics-wise :) $\endgroup$
    – kroko
    Oct 5 '20 at 15:11

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