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My primary tool is Interpolation: With[{poly = RandomPolygon[10], n = 50}, With[{len = ArcLength@RegionBoundary@poly}, (* Get a sequence of line segments. *) Partition[MeshPrimitives[poly, 2] …

This solution is a little bit... elaborate for the task, but shows how one can use GeometricTest to find solutions using synthetic geometry tools: With[ {c1 = {0, 0}, r1 = 1, c2 = {3, 4}, r2 = 4, …

I'm not entirely sure if your problem is fully defined, but here's one approach to start with, discretizing boundaries of finite components and showing them. CylindricalDecomposition is not limited to …

EDIT: I've updated the answer to use BinCounts, which suits this problem in multiple ways. Most of the old description below applies to this solution, but here the counts for intervals between (square …

You can find symbolic connected components (which are those regions you are asking about) in this case using CylindricalDecomposition. This can be a bit of an overkill if your goal is only to visualiz …

Since Mma v12.2 spatial point processes have opened yet another possibility with HardcorePointProcess, which prevents processes from resulting points being pairwise closer than a specified distance fr …

You can find a geometric solution to this problem by using an implicit region definition directly based on the definition of an ellipse: sum of distances from foci equals twice the semimajor axis. Wi …

I have the following problem, which can be stated geometrically as follows: List all pairs of vertex coordinates in a mesh region of measure 2 (a surface consisting of triangles) where these points a …

Assuming I understood your question correctly (there's really no need to use Module), here's one method which delegates calculating intersections to Solve: ClearAll@circleChord; circleChord[{xc_, yc …

Many solutions similar to how to get $n$ equidistributed points on the unit sphere are possible, especially if one can accept that points are not on the edges of a region. For instance, one can use an …

Mathematica 11.1 has a built-in function for this task: SpherePoints. This roughly corresponds CirclePoints for three-dimensional case; thus I wouldn't rely on it to get any sort of randomness on the …

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 …

Based on your image, but not your statement on lengths of lines: With[{eqn = a - a x / (1 - a)}, Show[ Quiet@Plot[Table[eqn, {a, 0, 1, 1 / 20}], {x, 0, 1}, Evaluated -> True, AspectRatio -> Aut …

If you want to simply sample the intersection, you can do this: RandomPoint[ ImplicitRegion[ RegionMember[ RegionIntersection[Sphere[{0, 0, 0}, 1], Sphere[{1, 1, 1}, 1.5]], {x, y, …

Somewhat dumb method (for instance, every line has both two Disks and a StadiumShape overlapping), but it's not at least very complicated: hulls0 = ConvexHullMesh /@ RandomReal[{-10, 10}, {3, 1, 2}]; …