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Turning my comment into an answer since I guess it's really a sufficient solution to the task at hand... The second answer is also semantically correct, Line can represent a collection of lines. If y …

When you see Element in the variable section of a function such as Maximize, it always means you are dealing with geometric regions, although it is not necessarily immediately obvious. … Values in geometric regions are always vectors, even when that vector has a length of one. So it happens to be in the case of Interval, when it is interpreted to be a region. …

This answer combines my own code from the question and the idea in @Quantum_Oli's answer. It is already quite much faster than either, at least for polygons: With[{reg = Polygon@RandomReal[{0, 10}, { …

It is likely that this method doesn't work for more complicated regions for which Integrate[Boole[...], ...] construct fails to produce meaningful symbolic results. …

How correct results (preferably for arbitrary regions) could be produced more efficiently? …

RegionDisjoint[reg, Rectangle[{x, y}]]]]]}]] These methods are usable on other regions too, say reg = Annulus[RandomReal[{-1, 1}, 2], Accumulate@RandomReal[{2, 5}, 2]] …

In general, GeoPositions and Regions don't mix in Mathematica, and if something like what's mentioned in the quesion works, it's a pure coincidence. … EDIT 2: Actually you can convert these regions which have been operated under a projection back to geographic computation entities, at least if the geometry is not too complex for MeshPrimitives (basically …

Given the input polygon is in polygon (with Polygon head), this hack finds approximate lines aligning with cylinders, identifies caps in a bit of an ad hoc manner, and then identifies corresponding cy …

Since v10.2 RandomPoint has provided a way to pick uniform samples from geometric regions (which you can trivially derive from your specification using ImplicitRegion): Eta[a_] := {Cos[a], Sin[a]}; NI … It is not particularly hard to construct regions where that doesn't apply. Thankfully it does work on most of simple cases one might explore... …

Element treats Intervals as geometric regions, and members of those geometric regions are vectors, even when they are of single dimension. … The fact that there are two different interpretations of an Interval - the old, and the new bought by geometric regions functionality - and the fact they're inherently single-dimensional objects causes …

This is a variation of @Syed's answer. Distance computation of convex hull boundary points is improved by using DistanceMatrix - which is very well optimised, even if half of the matrix is unnecessary …

EDIT: As @nikie noted, using FindArgMin (a variant of FindMinimum) instead of (N)ArgMin can improve the speed of finding a solution. Since in the case of this problem only one minimum exists, this sho …

Assuming polygons follow the same (clockwise or counterclockwise) vertex order, find all good quality two line segment rigid mappings between polygons without overlap with each other (at least much ov …

This answer is not so much about controlling the quality, but pointing out that Mma v13 introduces constructive solid geometry for some basic primitives such as Balls, and is capable of producing much …

Since discretisation is just numerics the more convoluted implicit regions don't really matter much here. … How these regions are split depend on the orientation of the region, although it's only a difference in rotation... …