If you put a different fork on a bike it no longer has the same angles .... Obviously. However, that is Not what is being discussed.

What is being discussed is being able to calculate stack and reach on a given frame when only the usual geometry charts are published---many of which do not include rake and trail info.

I am responding o the question which was asked.

obviously, fi the owner puts a 24-inch wheel in place of a 700c wheel, if the owner adds a suspension fork, if the owner adds a chopper-style springer front fork .... but what was being discuses was being able to calculate stack and reach given a common geometry chart. The poster stipulated that he knew the head- and seat-tube angles --- and given those angles, is implicit the horizontal plane or reference.

I know you are a serious mathematician, but I also know that while I can barley do 4-function math, I have done engineering drawings based off the geometry numbers listed, and have correctly calculated frame sizes.

Really, I think what the OP is saying is that given certain information, can certain other values be calculated. I say yes, having done so.

You are saying, if those dimensions are varied .....well, yes, change any of the lengths or angles and everything else changes. No one is arguing that. But that is not what the OP asked.

So long as the owner knows the seat- and head-tube angles, the owner know what "horizontal" and "vertical" are for that set of measurements, because the seat- and head-tube angles are measured relative the those "horizontal" and "vertical' planes. Those are industry standards---that is Exactly what those numbers mean.

Would you disagree? Is there some way of measuring those tube angles that I don't know about? (Well, no.)

I am not arguing that changing things does not change things. I am saying that none of that is what the OP asked, and none of it helps him with what he did ask.