Steinwurf mit Flatterband, Brücke mit Lampen

Idee Jörg Meyer, Hameln 10/99, Aufgegriffen Hellberg 02/01, 
Durchführung: Dörte Haftendorn 02/01

x[t_] := 
      t - (L/(Sqrt[1 + 4  (t^(2 ))]));y[
        t_] := -(t^2) - ((-2 L t)/(Sqrt[1 + 4  (t^(2 ))]));

Herleitung unten

Needs["Graphics`master`"]

L = 4;

bänder = 
        Table[Line[{{t, -(t^2)}, {x[t], y[t]}}], {t, -3, 3, 
            0.2}];
\(\(ParametricPlot[{{t, \(-t\^2\)} (*\(,\)\({x[t], 
              y[t]}\)*) }, {t, \(-3\), 3}, PlotStyle -> Farbig, 
        Epilog -> {Thickness[0.01], 
	
            Line[{{t, \(-t\^2\)}, {x[t], y[t]}}] /. t -> \(-1\), 
            Line[{{t, \(-t\^2\)}, {x[t], y[t]}}] /. t -> \(-0.2\), 
            Line[{{t, \(-t\^2\)}, {x[t], y[t]}}] /. t -> 0.2	, 
            Line[{{t, \(-t\^2\)}, {x[t], y[t]}}] /. 
              t -> 
                2, 
Line[{{t, \(-t\^2\)}, {x[t], y[t]}}] /. 
              t -> 1	, PointSize[0.03], 
            Point[{x[t], y[t]}] /. t -> \(-1\), 
            Point[{x[t], y[t]}] /. 
              t -> 2, 
Point[{x[t], y[t]}] /. t -> \(-1\), 
            Point[{x[t], y[t]}] /. t -> 1, 
            Point[{x[t], y[t]}] /. t -> \(-0.2\), 
            Point[{x[t], y[t]}] /. t ->  .2}
, 
        PlotRange -> {{\(-4\), 3}, {\(-6\), 3}}, 
        AspectRatio -> Automatic];\)
\)

L = 4; 
    bänder = 
      Table[Line[{{t, -(t^2)}, {x[t], y[t]}}], {t, -3, 3, 
          0.2}];
ParametricPlot[{{t, -(t^2)}, {x[t], y[t]}}, {t, -3, 3}, 
        PlotStyle -> Farbig, Epilog -> {bänder}, 
        AspectRatio -> Automatic];

\(\(\(Do[
      bänder = 
        Table[Line[{{t, \(-t\^2\)}, {x[t], y[t]}}], {t, \(-3\), 3, 0.2}]; 
	
      ParametricPlot[{{t, \(-t\^2\)}, {x[t], y[t]}}, {t, \(-3\), 3}, 
        PlotStyle -> 
          Farbig, 
		PlotRange -> {{\(-5.2\), 4}, {\(-13\), 
              5}}, 
				Epilog -> {bänder}, 
        AspectRatio -> Automatic], 
	{L,  .5, 
        5,  .25}]\)\(
\)\(	\)\)\)

ParametricPlot[{{t - 
              (4/(Sqrt[1 + 4 (t^2)])), -(t^2) + ((8  t)/(Sqrt[1 + 4 (t^2)]))}, {t, -2 (t^2)}}, {t, -2, 3}, 
AspectRatio -> 
          Automatic, PlotRange -> {-6, 3}];

Herleitung  ⇑

L =. ; x =. ; y =. ; u =. ; v =. ; t =.

Länge = ( (((x - u))^2) + (((y - v))^2) == 
          (L^2)); 
    Richtung = (((y - v)/(x - u)) == -2 t);

Länge = 
      Länge /. {u -> t, 
          v -> -(t^2)}
Richtung = Richtung /. {u -> t, v -> -(t^2)}

Solve[{Länge, Richtung}, {x, y}] // 
        Simplify;

{x -> (((t + 4 (t^3) - L (Sqrt[1 + 4 (t^2)]))/(1 + 4 (t^2))) // 
            Apart), 
      y -> (t (-t + ((2 L)/(Sqrt[1 + 4 (t^2)]))) // 
            Expand)}