# Photometry and seemingly two different definitions for foot-lamberts, why?

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". :)

• 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! Oct 28, 2020 at 0:54
• 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. Oct 28, 2020 at 0:58