Here's the link to the thread that was referred to.
Too much of it is in greek for me to understand if it explains the equation. I'll take a look at it later.
I can not entirely agree with it either.
Because, what usually would be the outer rim of the spill, is reflected towards the hot spot = throw.
If the hotspot gets much wider, there must be some focussing issues, which can be resolved with washers and / or a file.
Yet, a taller / deeper reflector is a scaled down version of a shallow one, but with more hight, so it ends up the same diameter as the short one. (I know what I mean, but will you…? 
)
So the emitter would be relatively bigger in a tall reflector, flooding the hotspot somewhat more than with a shorter / shallower one.
Yes, I guess the logic danger remains until you start experimenting yourself.
If the two given examples were not sufficient, you might suggest one. All you need is LED type (and flux-bin), current, and outer reflector diameter (actual value, not the bigger bezel). If you like you can even include the loss of the lens (w or w/o AR?) and the central hole diameter.
- The calculation requires led luminance (cd/mm²). A good start to determine it is led type and current in connection with the mentioned Leuchtdichten | Taschenlampen Forum, unfortunately not in english, yet. But with this hint you’ll get the idea with google translate. Luminance is the key to max-lux calculation. Unfortunately data sheets almost never list it. That’s the very reason why that page exists at all.
- The calculation also requires apparent reflector area. That’s the size of a flat disk directly in front of the reflector which would cover the bright parts of the reflector if you looked at it from the point of the spot. The actual paraboloid surface is not of concern (that’s why I chimed in, I hope it helps to point this out.)
The result of the calculation is luminous intensity, which is equivalent to the calculated lux at one meter. Actual measurements yield different results at one meter for throwers, because the light hasn’t converged yet.
LED size and reflector depth do not matter for max-lux in the spot center. I know that does sound odd at first.
Don’t forget about Dr. Jones excellent summary http://lux.yi.org/throw/, especially “What throw depends on”.
A deeper reflector will collect more lumens indeed, but it’s not the very center (max-lux) which benefits from this, but the outer parts of the spot area. Rather, the spot will become more “usable” by this. But now we speak about the beam profile which is a completely different animal. The above formula is only about max-lux, which could be a rather small center for a flat reflector.
Someone should give this program a shot.
~$7k to buy though. For that price I could probably write a program that would provide me with satisfactory results.
It’s logically safe to say that the same shape reflector, scaled larger will give more throw.
Almost everything else about flashlight design can also be reduced to mathematics, but that is a lot more work.
What flashlight has the longest known throw? That would be 622Kcd that rdrfronty measured from the Vinh modded TK61vn (hope I’ve got that model number right) So we can safely assume that the reflector on that light is the optimum design.
We can not assume that A + B = C as theres so many factors in play eg. driver, heatpath,etc.
A - TK61Vn
B - Longest known throw
C - Reflector with optimum design
interesting discussion though.
TK61 proves it - size does matter 
I resistor modded one (not as high as vinh does) with stock domed emitter and measured 265 kcd at 5 meters. To me it looks like a HD2010 reflector scaled up, pretty large flat area at the base of the reflector, around the emitter.
It is probably already said somewhere above in this thread, but:
Rules for smooth perfect paraboloid reflectors, as usual you do not get around physics (you can pray though ;-) ):
*same diameter=same throw
*two reflectors of same diameter: the deepest one will have the bigger hotspot, the shallowest one will have wider spill, but throw (brightness of the hotspot)=still equal
(But in practice smooth reflectors are hardly ever completely smooth, there are rings and other irregularities. Some companies make very good reflectors (I like Thrunite for that). And in practice emitters are often not centered, and out of focus)
…
Has anyone compared that to the monster-works of some of our elder statesmen here? I’m thinking ma_sha1, Match, texaspyro, (and too many others to enumerate) and even you, Dale, etc. might have a thing or two to say about that…
Just asking…
Here: https://omglumens.com/DEFT.php, at 1,000 kcd from Michael, same guy who has onestopthrowshop.com. jmpaul320 has one of these Deft-X lights.
There are other monster custom aspherics (maybe 700+ kcd), and more main stream, is the TK61vn (Fenix TK61 modded with high amps and de-domed XM-L2) that does 622 kcd - rdrfronty owns one and did the "official" throw measurement for vinh.
I'm sure many other modders here are capable of modding a TK61 the same way, pretty much, and get the same or close to 620 kcd out of it. vinh is doing some batch wide hand selection of emitters now, along with resistor modding, focusing adjestments, etc., same as many of us do.
Sure looks like Michael is claiming the world record for LED based flashlights with his DEFT-X.
Edit: here: http://www.peakbeam.com/. Max range of over 2 miles, yikes!
If the definition of "flashlight" is a hand help portable light source, than the Maxa Beam may be the best thrower available, manufactured that is...
That would be a telescope with a light source at its focus. Probably a space telescope such as the Hubble Telescope or a military surveillance telescope would throw farther than an Earth based one. I heard once that they bounced a laser off the moon.
(edit) Just to verify:
I think the thing is that an LED is not a point source.
If it would be a point source, diameter would theoretically not matter, only the ratio between width and depth.
To get close to the effect of a point source, the emitter should be as small as possible, in relation to the starting width (at 45° angle, which gets to reflect the warm coloured ‘side light’)
In other words, get a reflector that starts pretty wide, with a large ‘back hole’ (in relation to the LED)
Also decapitating the LED will help, because the dome magnifies the chip.
Still, when the parabolic (as close to it as you can get) reflector is also deep, in relation to the width, it should increase throw, because it throws more light in one direction, a higher percentage of the light emitted by the LED than when you cut off a portion of the mouth / flare / rim of the same reflector.
But that’s duh-obvious…
How would a reflector-flashlight do when you (somehow) also put a lens in it to bundle only the spill?
(I have some ideas on how to do that, but it’s not easy…)
Anyone try that yet?
No more black hole, like a TIR.
So where are the large TIRs?
I have so many questions…
Yes, a deeper reflector does catch more light (i.e. has smaller spill and more light in the collimated beam from the reflector), but no, it doesn't increase throw, instead it produces a wider corona, because you get smaller focal lengths. (Well, actually a deeper reflector does increase the throw a little, but only due to the smaller dead hole).
Also dedoming a LED does not increase throw because the dome enlarges the LED, but because of total inner reflexion and "photon recycling".
Large TIRs: these would be heavy pieces of massive material. Also the surface precision has to be better with larger optics.
My suggestion for even more throw (with a given diameter): Take several blue laser pointers and point them to the die to increase it's luminance.
But in case anyone has forgotten the point here, the question of this thread is REFLECTOR Width vs. Depth for Throw. Aspherics need not apply. Nor should telescopes or lasers or off Earth devices powered by NASA. The question is about flashlights with reflectors. The question is what combination of width/depth gives a reflector the best throw. Period.
Michaels DeftX looks sweet, it’s an aspheric, and I’d have to see it throw like he claims to believe it, as it’s an XP-G emitter. There are rules about the transmission of light. 4 times the light to travel twice the distance. The emitter has to be able to put forth sheer power. And we have all seen the XP-G footprint fall short. I can’t say it isn’t so, but I can so I don’t believe it without seeing it.
And still, the furthest throw document here is from that TK61vn. Anybody got one doing better than 622Kcd? Anyone? Measure the reflector on your best throwing light and see if it’s close to square…1:1 width vs depth. I’d like to know, really, I would. I’ve asked rdrfronty to take a look at that TK61vn. Give me some measurements.
And for the record. The driver, heat path, a whole bucket load of variables have little to do with how the reflector works to focus light from the emitter. A very nice reflector does indeed throw even Lo mode light surprisingly far. That being said, gotta have light to throw light. Load up a paint brush, make a stroke with it. How far did it go? Get a longer narrower brush, yada yada yada…you’re gonna be running out of paint in the brush regardless. Similarly, light can only travel so far without a whole lot more light pushing it. Or so it seems to me. But then, I’ve got a pretty good track record these days at being wrong….
I would like to verify again:
…because the parbole is relatively smaller with a deep reflector of the same width, so relatively the emitter is larger, not at all like a point source which would be perfect (with the perfect reflector) but more like a “cob LED” :^\quote].(Well, actually a deeper reflector does increase the throw a little, but only due to the smaller dead hole).Also dedoming a LED does not increase throw because the dome enlarges the LED, but because of total inner reflexion and “photon recycling”
[/quote]
My point would be that a domeless emitter is more like a point source than one with the dome, because with the dome, the light comes from a zone rather than a spot, scattering the light.
I’m not sure if PMMA is so heavy for making a, say, 40 or 50 mm diameter TIR.
That’s not on my to do list yet…
But yes, I have to do more stuff in practice and then talk about it… 
In the mean time I have a ‘lengtened C8’ on its way with an extremely deep reflector (60mm in stead of 30mm) and I already regret it… :Sp 
Dale - You are the one who asked "What flashlight has the longest known throw?" ?? Okkk, I'll forgot you asked , but I certainly did not forget the point here...
I have a TK61, belongs to DayLighter, but I resistor mod'ed it for him and still have it while working on another light for him.. I had it all apart - it had no glue, was easy to get access to the emitter, driver, etc. I can measure the reflector if you want. Like I said previously, it looks like a HD2010 reflector on a larger scale, roughly same proportion of width to depth. Also I posted on it in this thread on the TK61: https://budgetlightforum.com/t/-/23474.
Again, primary, main reason why the TK61vn is king of "reflector" throwers is because it's reflector's diameter is the largest, plus easily resistor modded to high amps. Of course other factors of design/shape - for example, though the NiWalker had a slightly wider diameter reflector than a TN31, it could not throw as well - it did good, but not as good as a TN31 - thinking the amps was also a factor - could not get the amps up as high as you can with a TN31.
Unfortunately many of us (me too! ) don't invest in the high end name brand lights to tinker around with. Vinh can of course because he makes a living on mods and has a large customer base that will pay big $$$.
If that's true,then dedoming color LEDs wouldn't increase throw.
That is the more fundamental reason, in that it applies to optimal optics, but the visible fact that it enlarges the led is also relevant. The dome narrows the angular spread, which reduces the throw of a reflector light.
I'd like to point to the dedoming thread where that is discussed, to avoid going off-topic too much.