Distance between polygons

Ideas
,
1

distPolygons[poly1_, poly2_] := {Sqrt[(x1 - x2)^2 + (y1 - y2)^2] /. #, #} & @
QuadraticOptimization[(x1 - x2)^2 + (y1 - y2)^2, {Element[{x1, y1},
poly1], Element[{x2, y2}, poly2]}, {x1, y1, x2, y2}]

 p1 = RegularPolygon[{0, 0}, {1, \[Pi]/7}, 3]
 p2 = RegularPolygon[{2, 3}, {1, -(\[Pi]/5)}, 4]
    
 sln = distPolygons[p1, p2]
    
{1.82996, {x1 -> 0.900969, y1 -> 0.433884, x2 -> 1.41221, 
      y2 -> 2.19098}}

polydist.jpg
polydist.jpg221×288 10.8 KB

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2

Nice! Here is a more lazy way to do it:

d12 = NMinimize[RegionDistance[p2, {x, y}], {x, y} \[Element] p1];
d21 = NMinimize[RegionDistance[p1, {x, y}], {x, y} \[Element] p2];
Graphics[{p1, p2, Red, Line[{{x, y} /. d12[[2]], {x, y} /. d21[[2]]}]}]
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3

Henrik,

Your method is much slower than mine. Since the objective function is quadratic and the constraints for a point to be in a polygon are linear, QuadraticOptimization is the best approach.

Frank

4

Frank,

Your method is much slower than mine.

yes. of course! Being lazy can be costly. Regards – Henrik

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