How to minimize the maximum aspect ratio of two subpolygons?

Question

I would like to cut a convex polygon into two with a given area ratio using a straight line, so that a larger aspect ratio of the two sub-polygons is minimized.

My approach at the moment involves choosing a random starting point, calculating the corresponding end point, which divides the polygon into target areas, and then calculates the larger of the two aspect ratios. Repeating this many times until I got to a minimum!

The aspect ratio of the polygon A is defined as:

asp(A) := diam(A)^2 / area(A)

Answer 1

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containsRatioQ[area1_, area2_, areaBetween_, ratio_] := 
 IntervalMemberQ[areaRatioInterval[area1, area2, areaBetween], ratio]

• paramater labda, mu (, )

  \[Lambda] -> (2*aL + givenAreaRatio*(-2* aR + (p1y - p3y)*(p2x - p4x) - (p1x - p3x)*(p2y - p4y)) + (1 + givenAreaRatio)*(p1x*p3y - p3y*p4x + p1y*(-p3x + p4x) - p1x*p4y + p3x*p4y)*\[Mu])/ ((1 + givenAreaRatio)*((-p2x)*p4y + p1x*(-p2y + p4y) + (p1x - p2x)*(p3y - p4y)*\[Mu] + p1y*(p2x + p4x*(-1 + \[Mu]) - p3x*\[Mu]) + p2y*(p4x + p3x*\[Mu] - p4x*\[Mu])))

mu labda:

\[Mu] -> (-2*aL + givenAreaRatio*(2*
       aR - (p1y - p3y)*(p2x - p4x) + (p1x - p3x)*(p2y - p4y)) + (1 + 
      givenAreaRatio)*((-p1x)*p2y + p1y*(p2x - p4x) + p2y*p4x + 
      p1x*p4y - p2x*p4y)*\[Lambda])/
   ((1 + givenAreaRatio)*((-p3y)*p4x + p3x*p4y + 
     p1y*(p3x - p4x)*(-1 + \[Lambda]) - 
     p1x*(p3y - p4y)*(-1 + \[Lambda]) + ((-p2y)*p3x + p2x*p3y + 
        p2y*p4x - p2x*p4y)*\[Lambda]))

p1, p2, p3, p4 , , aL - "" , aR - "" (. ).

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• . Mathematica:

  NMinimize[{Max[aspectRatio[area1tot], aspectRatio[area2tot]], \[Mu]range[[1]] <= \[Mu] <= \[Mu]range[[2]]}, \[Mu]]

are1tot area2tot , mu lambda, mu.

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Answer 2

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