# Probability distribution over shapes (or: How to parameterize arbitrary polygons)

Has there been work on modeling variations of a 2D shape? E.g., say you want a distribution over 5-sided polygons, or over ellipses, or curved shapes? For simple shapes, like circles, rectangles, etc., this seems really straightforward, but I'm having a hard time coming up with a reasonable way to model more complicated (especially curved) shapes. I suppose for curved shapes I could just treat them as polygons and then do spline interpolation. But even then I would need a distribution over arbitrary polygons. Any ideas?

Edit after comments: I guess the problem isn't one of a "distribution over polygons", but one of parameterizing polygons such that I can put distributions over the parameters. I still don't know how to do this for arbitrary (closed, non-self-intersecting) polygons.

Context: I would like something to sample from that will generate puzzle pieces (e.g., https://static.brusheezy.com/system/resources/previews/000/024/418/original/50-puzzle-pieces-brushes.jpg, https://www.libertypuzzles.com/userfiles/kcfinder/images/complex-whimsy-piece-300.jpg, https://www.mgcpuzzles.com/mgcpuzzles/images/all_new_core_images/jigsaw_puzzle_piece/traditional_puzzle_pc_ap_1T.jpg). This seems to boil down to parameterizing polygons. I could simplify things a bit by modeling the general outline of the piece, the location of the interconnects, and the shape of the interconnects separately, but there are still two polygons in there that need to be modeled (the general outline, and the shape of the interconnect).

• These are intended to be closed (as with polygons and ellipses), or is this more general? – Glen_b Jul 6 '18 at 0:21
• What is wrong with parameterizing the shape and use some probability distribution for the parameters? – Martijn Weterings Jul 6 '18 at 8:32
• If you need some specific distribution for a specific characteristic (or more) then you can express this characteristic in terms of the parameters and work out an appropriate distribution for the parameters. E.g for rectangles you might want all volumes between $a$ and $b$ to be equally possible. Then you could use a uniform distribution for volume = l x h and any distribution for the shape l/h. – Martijn Weterings Jul 6 '18 at 8:33
• @mattg actually I was asking for clarification, instead of providing it. The term 'probability distribution of shapes' is so general. It is unclear what you exactly mean. – Martijn Weterings Jul 6 '18 at 13:56
• Since it's difficult to characterize a proper polygon with coordinate-based formulas, your task is exceptionally broad and likely hopeless. Could you explain why you want to model random polygons and provide some information--any information--that would help people deduce some characteristics of their distribution? That might narrow your question enough to make it answerable. – whuber Jul 6 '18 at 14:13