Calculate largest rectangle in a rotated rectangle

Question

I\'m trying to find the best way to calculate the biggest (in area) rectangle which can be contained inside a rotated rectangle.

Some pictures should help (I hope) i

Answer 1

sorry for not giving a derivation here, but I solved this problem in Mathematica a few days ago and came up with the following procedure, which non-Mathematica folks should be able to read. If in doubt, please consult http://reference.wolfram.com/mathematica/guide/Mathematica.html

The procedure below returns the width and height for a rectangle with maximum area that fits into another rectangle of width w and height h that has been rotated by alpha.

CropRotatedDimensionsForMaxArea[{w_, h_}, alpha_] := 
With[
  {phi = Abs@Mod[alpha, Pi, -Pi/2]},
  Which[
   w == h, {w,h} Csc[phi + Pi/4]/Sqrt[2],
   w > h, 
     If[ Cos[2 phi]^2 < 1 - (h/w)^2, 
       h/2 {Csc[phi], Sec[phi]}, 
       Sec[2 phi] {w Cos[phi] - h Sin[phi], h Cos[phi] - w Sin[phi]}],
   w < h, 
     If[ Cos[2 phi]^2 < 1 - (w/h)^2, 
       w/2 {Sec[phi], Csc[phi]}, 
       Sec[2 phi] {w Cos[phi] - h Sin[phi], h Cos[phi] - w Sin[phi]}]
  ]
]