I agree with all of the above, and certainly feel grateful that I have a much better understanding now than before this discussion, despite the inability to reach an agreement. I was being overly negative in the previous post, and apologize for that.
That is very reasonable criticism, and it remains to be shown (beyond personal observation) that my definition of “hotspot” agrees with what most people would perceive as the hotspot.
Sadly it’s hard to find/measure real-world data for unusual reflector geometries (e.g., very wide or very narrow), but this simulation (which has been around for a long time) should give some insight. For more common geometries, maybe I could try reddit to get some data to test our models on.
I absolutely agree that the conservation of etendue is a real thing. It’s just that I don’t believe every derivation must be done through etendue calculations; instead, I believe that any correct derivation must produce a result that is consistent with conservation of etendue, even without using the principle. The same way that simple physics problems can be solved without appealing to conservation of energy or the second law of thermodynamics: any correct solution–no matter how it’s found–must be consistent with them.
I chose the ray-tracing route over the etendue route for my derivations simply because the math is easier. To use etendue in full effectiveness/generality involves not just multiplying emitter area and emitting angle, but extremely ugly integration. On the other hand, dealing with reflector geometry requires little more than basic trigonometry, and some occasional root-finding.