Orientierungstreue lineare Abbildungen

F = {{1, 0}, {1, 3}, {2, 3}, {2, 2.5}, {1.5, 2.5}, {1.5, 2}, {2, 
          2}, {2, 1.5}, {1.5, 1.5}, {1.5, 0}, {1, 0}}

achsenkreuz = Plot[{}, {x, -5, 5}]

Show[achsenkreuz, Graphics[Line[F]], 
      PlotRange -> {{-5, 5}, {-4, 4}}, 
      AspectRatio -> Automatic]

A = {{-1, -(1/4)}, {1, -(1/2)}}

Det[A]

F2 = Map[A . # &, F]

Show[achsenkreuz, Graphics[Line[F]], Graphics[Line[F2]], 
      PlotRange -> {{-5, 5}, {-4, 4}}, 
      AspectRatio -> Automatic]

B = {{-(1/4), 1}, {1, (1/2)}}

Det[B]

F3 = Map[B . # &, F]

Show[achsenkreuz, Graphics[Line[F]], Graphics[Line[F3]], 
      PlotRange -> {{-5, 5}, {-4, 4}}, 
      AspectRatio -> Automatic]

Drehungen und Spiegelungen

A = {{Cos[α], -Sin[α]}, {Sin[α], 
          Cos[α]}}

Det[A]

Det[A] // Simplify

F2 = Map[A . # &, F] /. {α -> (π/3)}

Show[achsenkreuz, Graphics[Line[F]], Graphics[Line[F2]], 
      PlotRange -> {{-5, 5}, {-4, 4}}, 
      AspectRatio -> Automatic]

Eigensystem[A]

A = {{Cos[α], 
          Sin[α]}, {Sin[α], -Cos[α]}}

Det[A]

Det[A] // Simplify

F2 = Map[A . # &, F] /. {α -> (π/3)}

Show[achsenkreuz, Graphics[Line[F]], Graphics[Line[F2]], 
      PlotRange -> {{-5, 5}, {-4, 4}}, 
      AspectRatio -> Automatic]

Eigensystem[A]