Intersecting regular polygons

[Dr George Whale]

The recent, nicely illustrated 'Overlapping Circles' post got me thinking about intersecting regular polygons, specifically the scenario in this demo:

https://openprocessing.org/@GeorgeWhaleResearch/2990346

(Set variables s (num. poly sides) and n at top.)

The demo uses a formula by Claude.ai (see bottom-left), but (as a non-mathematician)... can I trust it?

You'll see from the demo that intersection points get closer together as n increases, and intuitively I'd say points can never merge. But could any of the mathematicians here offer something more solid than intuition to back this up?

Also, you'll see that for, e.g., n = 3, 6, 9 etc. in the triangles case, there are multiple triangles in precisely the same position / orientation. Can one speak of intersections when a line / edge lies exactly on top of another? Claude says not - is that the convention?

Thanks in advance

George W.