(* This mathematica file creates the frames for three animations which
   animate a bouncing ball. The first animation only shows the ball
   together with a clock and axes. The second adds the graph of
   the height versus time, and the third displays the graph of the
   derivative as well.  (Actually we show a graph of the derivative
   divided by 20, so that the same axes can be used for the function and
   the derivative.

   The ball is released at time 0 from 100 centimeters.  We display
   its height versus time over the interval [0,3], measured in second,
   although it comes to rest at time 2.4 seconds.  The height function
   is a piecewise quadratic.  It is reasonably realistic, but not meant
   to conform to any exact model.

   Douglas N. Arnold, 8/29/92, 8/25/94

   Douglas N. Arnold  <dna@math.psu.edu>
   Department of Mathematics
   Penn State
   University Park, PA  16802

   Online documentation related to this file can be found at WWW URL
   http://www.math.psu.edu/dna/graphics.html.
*)

BeginPackage["Bounce`", "Graphics`Colors`"]

Begin["Private`"]

(* Define the ground contact times.  These have been adjusted to give a
fairly realistic bouncing ball, but also to be integers/20.  This way we
can animate with frames every 1/20 second and the ball always hits the
ground exactly at a frame time.
*)

xp = {-10, 10, 23, 32, 38, 42, 45, 47, 48}/20
g[x_]:= 0 /; x <= First[xp] || x >= Last[xp]
g[x_]:=
  Module[{m},
    m = Last[Position[Map[(#<x)&,xp],True]][[1]];
    Return[400(x-xp[[m]])(xp[[m+1]]-x)]; ]

(* Define its derivative divided by 20 *)

gp[x_]:= 0 /; x <= First[xp] || x >= Last[xp]
gp[x_]:=
  Module[{m},
    m = Last[Position[Map[(#<x)&,xp],True]][[1]];
    Return[20(xp[[m]]+xp[[m+1]]-2 x)]; ]

(* Set limits of display, times for frames, and options to ShowAnimation. *)

    {{xmin,xmax},{ymin,ymax}} = {{-.5,3},{-20,120}}
    times = Table[t,{t,0,3,1/20}]
    opts0 = ( AnimationFunction -> (DisplayAnimation["bounce0.anim",#]&) )
    opts1 = ( AnimationFunction -> (DisplayAnimation["bounce1.anim",#]&) )
    opts2 = ( AnimationFunction -> (DisplayAnimation["bounce2.anim",#]&) )

(* Define a single frame of the animation. *)

frame[i_,showgraph_,showdgraph_] :=
        ( x = N[times[[i]]];
        y = g[x];
(* Create the graph, a polyline plotting the height of the ball versus time
   from time 0 to the present moment.  Use the points with integer abscissas
   as breakpoints for the polyline except include the present point as
   well and include the starting point twice so that Line doesn't get
   confused on the first frame when it has only one endpoint. *)
        graph = If[showgraph,
         { Blue, Thickness[0.007],
             Line[Map[{#,g[#]}& , Prepend[Union[Table[t,{t,0,x,1/20}],{x}],0]]]},
     {} ] ;
(* Create the graph of the derivative.  This is a bit of a bother since
   the derivative is undefined at the break points.  *)
   dgraph = If[showdgraph,
     eps = .005;
     m = Last[Position[Map[(#<x)&,xp],True]][[1]];
     Flatten[{Red, Thickness[0.007],
     Table[Line[Map[{#,gp[#]}&,{Max[xp[[j]],0]+eps,xp[[j+1]]-eps}]],{j,1,m-1}] ,
     Line[Map[{#,gp[#]}&,{Max[xp[[m]],0]+eps,Max[x,2 eps]-eps}]] }],
     {} ] ;
(* Create axes for the graph. *)
	axes = {Black, Thickness[0.001], Line[{{0,-20},{0,120}}],
	        Line[{{0,0},{60,0}}]};
(* Create a solid rectangle for the ball to bounce on. *)
	rectangle = {Black, Polygon[{{-.5,-20},{0,-20},{0,-3},{-.5,-3}}]};
(* Create a bouncing ball. *)
	ball = {Pink, PointSize[0.03], Point[{-.25,y}]};
	dot = {Black, PointSize[0.005], Point[{-.25,y}]};
(* Create the clock. *)
        xc = xmin + .75 (xmax-xmin);
	yc = ymin + .75 (ymax-ymin);
	xr = .1 (xmax-xmin);
	yr = .15 (ymax-ymin);
	border = {Goldenrod, Thickness[0.001],
	Polygon[Table[{xc,yc}+ {xr Cos[th],yr Sin[th]},{th,0,Pi, Pi/25}]]};
        base = {Goldenrod, Thickness[0.001],
	Polygon[{{xc-xr,yc},{xc-xr,yc-yr/4},{xc+xr,yc-yr/4},{xc+xr,yc} }]};
	minordots = Flatten[{Black,PointSize[0.006],
	Table[Point[{xc,yc}+ {xr Cos[th],yr Sin[th]}],{th,0,Pi,Pi/30}]}];
	majordots = Flatten[{Black,PointSize[0.012],
	Table[Point[{xc,yc}+ {xr Cos[th],yr Sin[th]}],{th,0,Pi,Pi/6}]}];
	centerdot = {Black,PointSize[0.02],Point[{xc,yc}]};
	numbers = {Text[0,{xc-1.2 xr,yc}],
                   Text[1,{xc-.6 xr,yc+.6 Sqrt[3]yr}],
                   Text[2,{xc+.6 xr,yc+.6 Sqrt[3]yr}],
                   Text[3,{xc+1.2 xr,yc}]};
	clockhand = {Black,Thickness[0.005],
               Line[{{xc,yc},{xc,yc}
                   +.85 {xr Cos[Pi-Pi x/3],yr Sin[Pi-Pi x/3]}}]};
	clock =
         Flatten[{border,base,majordots,minordots,centerdot,numbers,clockhand}];
(* Combine everything in a Graphics object. *)
        Graphics[Flatten[{ graph, dgraph, axes, rectangle, ball, dot, clock }],
          PlotRange->{{xmin,xmax},{ymin,ymax}}, Frame->True] )

(* Create the frames. *)

prefix = "bounce0/T";
pixels=400;
filename[prefix_,number_] := StringJoin[prefix,ToString[number],".gif"]
Table[ Display[filename[prefix,i],frame[i,False,False],"gif",ImageSize->pixels],
  {i,1,Length[times]}]
prefix = "bounce1/T";
filename[prefix_,number_] := StringJoin[prefix,ToString[number],".gif"]
Table[ Display[filename[prefix,i],frame[i,True,False],"gif",ImageSize->pixels],
  {i,1,Length[times]}]
prefix = "bounce2/Tounce0/T";
pix";
filename[prefix_,number_] := StringJoin[prefix,ToString[number],".gif"]
Table[ Display[filename[prefix,i],frame[i,True,True],"gif",ImageSize->pixels],
  {i,1,Length[times]}]

End[] (* end of context private *)

EndPackage[]

(* write an exit message *)

Print["-- wrote frames --"]