I’m afraid that the -1 term breaks the proportionality–please check your calculations.
The alternative 0.5log_2 (4pi cd/lm) achieves the same normalizations, and so do many expressions. None of them is particularly justified over any of the others–at this point it becomes more numerology than mathematics.
@sb56637 Thank you for separating our side-discussion out of the convoy thread into its own!
Agreed. And yet it is the most underrated category, as it is too throwy for most people’s EDC and not throwy enough for a pocket thrower. It is however the true do-all a-ok beam shape.
I misspoke and I stand corrected. The √(cd/lm) is however proportional to throw and nearly inversely proportional to cone apex planar angle - both common measures of beam shape already in use. Other transformations of cd/lm will not have this property, but they can be forced to pass through the ‘anchor’ points as you have done with your log expression.
I also argue that this type of beam shape index is not that far from being proportional to perceived brightness of the target illuminated by the beam. As we say in Astrology: the stars aligned :-)
I’m afraid it’s more complicated. The just noticable difference has been shown to be rather scale dependent and Benford’s law describes the frequency of leading digits of random values spanning large ranges and I don’t think applies here.
This paper discusses it well. Both Steven’s and Weber-Fechner’s are probably wrong, and just useful as first-order empirical approximations, with Steven’s having an advantage of being successfully applied in image processing under the disguise of gamma. Nature doesn’t seem to care much about the neatness of equations describing it or historically important Weber’s insights and assumptions :-)
Technically correct, but it may be the case of mathematical exactness obscuring the forest for the trees. :-)
If you drop 1, the scale’s zero will represent the beam shape of a Mule (a reasonable starting point for a descriptor of flashlight beams) rather than the omnidirectional source. With 1, it’s still linear and increasingly closer to proportional for narrower beams, but with zero (not -1) being the absolute minimum or no ‘beam’ at all - either one may be super-good-enough.
This is a good argument for sqrt(cd/lm) being a reasonable metric. To clarify for fellow readers: “throw” means “ANSI distance”, and “cone apex planar angle” means divergence (diametrical angle) of the beam.
Folks who prefer to use ANSI m may prefer your sqrt(cd/lm), while folks who prefer to use lux @ 1m (equivalently, candela) would prefer just cd/lm.
Very interesting! BTW, the essence of Benford’s law is not about the first digit, but that quantities spanning many orders of magnitude tend to be log-normally distributed, rather than normally, suggesting that log is the right scale. The first digit thing is the symptom, not the underlying cause.
I will be checking out the paper–thank you for sharing it.
When someone makes a mathematically precise statement, I expect it to be correct. No forests and no trees in this picture.
If they would like to state some approximate idea that does not require exactness, they should present it as such.
If you drop the -1 term, the remaining formula can never equal zero. Could you expand on what you mean?
I don’t quite see the purpose of finding a difficult-to justify transformation, when cd/lm (or its sqrt) is already easily interpretable and enjoys linear mathematical relations with other commonly-used quantities.
Any nontrivial transformation would wreck havoc on these nice linear relations and make computation/conversion between units a nightmare for folks who are not well-versed in mathematics: just look at the masses of reddit comments who think the inverse-square law in physics is a perception law but are unable to even describe what it says. Not to mention that one often cannot assign well-defined units to the transformed quantity…
True. Wrong direction. -2 would bring the anchor points mapping to -1 and zero. Dropping the term will map the anchors to 1 and 2.
The only reason for embellishing the √(cd/lm) was to map the physically based beam shape limits of 1/4π and 1/π into recognizable round index values. It’s good to go without it, but is harder to interpret in absolute terms. Though in my case it turns out that straight √(cd/lm) happens to be about 1 for HS21 Flood and 3 for Spot which makes it easy (for me) to interpret other beam shapes in comparison.
Just for fun, if BS = √(cd/lm) - for Beam Shape… - then to get approx. corresponding beam divergence angle in degrees:
Angle ≈ 65/BS
To get the ANSI m:
Throw = 2BS×√lm
For instance the BS involved in T6 SFT25R is roughly 6, which translates to about 11° beam and some 460 m throw (@1500 lm).
3×21D is afflicted with the BS of about 12 which corresponds to some 5° beam angle and ~1.6 km throw (@4500 lm).
Relative comparisons are even easier: all other things staying constant, double BS implies double throw and close to half the beam angle (and, in the realm of the senses, I’m convinced it makes the target about twice as bright, or there abouts, as well).
Very nice! This approximation holds for throwy beams and follows from a small-angle approximation of
where C = cd/lm.
The precise constant for the approximation is 360/π^(3/2), which comes out to around 64.65.
This is indeed an exact equality as the expression simplifies to 2sqrt(cd), which is the usual definition of throw.
On the other hand, it implies quadrupling of intensity and quartering of illuminated area. Whether to work with additive area-based units (lux or cd) or Pythagorean-additive length-based units (ANSI m) is entirely up to personal preference.
Very nice!
I like this approximation for the beam angle. This might be the most impactful discovery in this thread.
This makes it simple to give a rough estimate of the beam angle just based on cd and lumen measurements which are given in each flashlight review.
This does however not factor in that only a part of an LED’s emitted light is collected into the hotspot. This is thus overestimating the beam angle. It should likely be reduced by a factor of sqrt(h), where h is the fraction of lumens collimated into the hotspot, likely ~0.5-0.7.
Good point. It’s probably easier not to make the hot spot beam/spill contribution corrections and treat the (max)cd/(total)lm as a ‘nominal’ beam shape characteristic for easy ranking of different lights.
There are other implicit assumptions here, including the beam being a sharp-boundary cone of uniform intensity as measured by cd-max.
I can think of a problem though if one wants to estimate the cd/lm factor (or beam angle) geometrically from direct hot spot diameter vs. distance measurements. If half the lumens go into hot spot beam and the other half to spill, the beam angle derived from hotspot may be some 40% tighter than the ‘nominal’ one using max_cd/lm.
Yes, the nominal/upper bound angle approximation could be good enough, except for two issues:
First, it will deviate significantly from correct measures, such as Armyteks.
Second, it is only proportionally correct for similar optics. A shallow or deep reflector will differ in how much light is collected, and TIRs differ even more.
It seems like some correction factors would need to be estimated and applied for this measure to be truly useful.
I treat it as the flashlight being a black box - its light source, by some miracle, produces X lumens of light total. Then the torch concentrates it into Y cd intensity beam that I want. cd/lm and its derivatives measure how efficiently the flashlight does it, spill be damned (it’s the flashlight’s problem what it does with all these lumens - I just want the beam).
I kind of understand the BS = √(cd/lm) scale more now - my lights vary from 1 to 12 on this scale. Keeping it nominal sidesteps the issues of beam non-uniformity, fuzzy boundary cut off points (FWHM?), and subtracting spill contributions, while still describing the beam shape (or more like how well the flashlight concentrates given amount of light into a beam) on an intuitively equidistant scale.
Do you know how armytek does its beam angle measurements? They report something like hotspot diameter at a fixed distance?
Another question: are you aware of measurements that could elucidate what portion of light fuels the beam and what goes to spill for different optics configurations?
Nominal/effective ratings are useful in exactly this way. Most Li-ion cells are rated as 3.7V nominal, yet they do not agree with this voltage for the vast majority of the discharge–it’s always higher or lower. However, it is a very useful for making “total energy” (mWh) calculations because it is a good approximation to the average voltage across a constant-current discharge cycle.
Very good questions. Reflectors are simple enough that if you know the angle of the opening with vertex at the emitter, you can figure out the proportion of light going into the spill, and into hotspot+corona. It is more difficult to separate hotspot and corona (since both come from light incident to the reflector), but calculations can be done, in principle at the very least.
Aspherics/Fresnels are even simpler, having just a hotspot and no spill.
With TIR’s, it’s truly the wild west and no nice formula can exist because the geometry of the optic is so different across different models.
I must respectfully disagree. It’s going to be a compound formula, similar to a grand unified theory. At the very least, you will need to incorporate the diameter, outer angle, the adjacent inner angle, the inner LED adjacent inner diameter, the curve of the LED adjacent aspheric, the curve of the emitting surface aspheric, the inner diameter of the emitting aspheric, overall height, maybe focal lengths, and all corresponding angles of incidence.
Most TIR will have some aspect shared across TIR, with some sloping all the way to the bottom, and some sloping to a pedastal. Just because it seems chaotic doesn’t mean it is, chaos is merely the illusion caused by ignorance or unaccounting.
Sure, but to what end?
I’m not patient enough (or these days, intelligent enough) to figure it out, but all the pieces are there.
It seems that you are arguing for my point, rather than against it!
There are still a few very important issues you left out: for starters, many TIRs don’t even have a well-defined spot/spill cutoff.
In addition, many TIRs don’t even have a parabolic surface, so describing the lateral surface itself requires possibly infinitely many parameters. Furthermore, some TIRs incorporate also a convex lens, some don’t; some have diffusion, some don’t. Some TIRs have a two-part design where the lateral surface is glued together from two parabolas with different curvature (e.g., Convoy’s narrow 3535 and 5050 optics), producing a two-layer hotspot. How can you possibly account for all these variations in one formula?
Can’t agree or disagree without knowing what your definition of chaos is.
Read up on the Abel-Ruffini theorem, which (roughly) states that there is no analogue of the quadratic formula for polynomial equations of degree 5 or above. Not that humans are not smart enough to find such a formula, but that one cannot possibly exist.
On this general topic, check out what @QReciprocity42 wrote here: Wurkkos TS30S + SBT90.2 monster = $60 4750lm 1km SUPER-Thrower. Comparison w/ other big time throwers: TS30S Pro, Firefly T9R w/ FFL909MX, & INCREDIBLY 😈 : Convoy M21E with SFT-25R 5000k! (Summary & measurements on P. 1) - #671 by QReciprocity42
Upon given 2 separate equations and asked how one could possibly become the other in middle school, I inadvertently rediscovered l’hopitale’s formula for derivations, only, I didn’t know what it was or even recognize it until I studied calc in high school. I just thought it was cool that I was one of two to figure out how it was possible, only I learned later that the other guy had already been taught how to do it. And by my own standards, I was already kinda dumb back then…
That said, there’s loads more people more intelligent than I am, so surely out of 8 billion people on the planet, one of them will rediscover how a TIR simulation program actually renders. 
Repost this informative post here so I could look at it again when blood flow to my brain is low.