(*^

::[	frontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.1";
	macintoshStandardFontEncoding; 
	paletteColors = 128; automaticGrouping; currentKernel; 
	fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8,  24, "Times"; ;
	fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6,  18, "Times"; ;
	fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6,  14, "Times"; ;
	fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20,  18, "Times"; ;
	fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15,  14, "Times"; ;
	fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12,  12, "Times"; ;
	fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; ;
	fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  10, "Times"; ;
	fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-5,  12, "Courier"; ;
	fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5,  12, "Courier"; ;
	fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5,  12, "Courier"; ;
	fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5,  12, "Courier"; ;
	fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5,  12, "Courier"; ;
	fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287,  12, "Courier"; ;
	fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic,  10, "Geneva"; ;
	fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; ;
	fontset = leftheader, inactive, L2,  12, "Times"; ;
	fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7,  12, "Times"; ;
	fontset = leftfooter, inactive, L2,  12, "Times"; ;
	fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  10, "Times"; ;
	fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; ;
	fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; ;
	fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; ;
	fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; ;
	fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; ;
	fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; ;
	fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; ;
]
:[font = special1; inactive; preserveAspect; ]
                                                         Virtual Reality -- Flatland
  
The closed initialization cell below defines several Mathematica  procedures for working with images.

Each image is represented as a list of points -- written

                                              {{x1, y1}, {x2, y2}, ... {xn, yn}}

The image is drawn by connecting the points in order.

The initialization cell defines the following procedures.

ShowCurve[list, n]

This procedure displays an image of a curve and requires two arguments.  The first argument is a list representing a curve as described above.  The second argument, n, sets the scale for the x- and y-axies.  The two axes go from -n to n.

ShowCurves[list, n]

This procedure is similar to the first one except it displays a list of curves.

Mirror[list]

This procedure  reflects an image represented by a list in the x-axis.

Rot[list, theta]

This procedure rotates an image represented by a list around the origin by an angle of theta radians counterclockwise.

Trans[list, a, b]

This procedure translates an image represented by a list a units horizontally and b units vertically.

Dilate[list, factor]

This procedure dilates an image represented by a list by the factor, factor.

Ident[list]

This procedure leaves an image alone.
;[s]
35:0,0;57,1;84,0;141,2;152,0;295,1;329,0;445,1;463,0;630,1;631,0;694,1;696,0;700,1;701,0;704,1;723,0;806,1;818,0;892,1;908,0;997,1;1002,0;1030,1;1047,0;1106,1;1107,0;1131,1;1132,0;1152,1;1172,0;1243,1;1249,0;1252,1;1263,0;1303,-1;
3:18,13,9,Times,0,12,0,0,0;16,13,9,Times,1,12,0,0,0;1,13,9,Times,2,12,0,0,0;
:[font = input; initialization; closed; preserveAspect; ]
*)
Clear[ShowPoint, ShowPoints, ShowCurve, ShowCurves,
      Transpt, Trans, Mirrorpt, Mirror, Rotpt, Rot,
      Identpt, Ident]

ShowPoint[pt_, high_:8] := 
    Show[Graphics[{PointSize[0.02], Point[pt]}],
    PlotRange -> {{-high, high}, {-high, high}},
    AspectRatio -> Automatic,
    Frame -> True]
    
ShowPoints[pts_, high_:8] := 
    Show[Graphics[{PointSize[0.02], 
    Map[Point, pts]
    }],
    PlotRange -> {{-high, high}, {-high, high}},
    AspectRatio -> Automatic,
    Frame -> True]
    
ShowCurve[curve_, high_:8] :=
    Show[Graphics[Line[curve]],
    PlotRange -> {{-high, high}, {-high, high}},
    AspectRatio -> Automatic,
    Frame -> True]

ShowCurves[curves_, high_:8] :=
    Show[Graphics[{PointSize[0.3],
    Map[Line, curves]}],
    PlotRange -> {{-high, high}, {-high, high}},
    AspectRatio -> Automatic,
    Frame -> True]    

Transpt[pt_, a_, b_] := {pt[[1]] + a, pt[[2]] + b}

Mirrorpt[pt_] := {pt[[1]], -pt[[2]]}

Rotpt[pt_, theta_] :=
{pt[[1]] Cos[theta] - pt[[2]] Sin[theta],
 pt[[1]] Sin[theta] + pt[[2]] Cos[theta]}
 
Dilatept[pt_, ftr_] := {pt[[1]] * ftr, pt[[2]] * ftr}
 
Identpt[pt_] := pt

Trans[lst_, a_, b_] := 
  Table[Transpt[lst[[k]], a, b], {k, 1, Length[lst]}]
  
Mirror[lst_] := 
  Table[Mirrorpt[lst[[k]]], {k, 1, Length[lst]}]
  
Rot[lst_, theta_] := 
  Table[Rotpt[lst[[k]], theta], {k, 1, Length[lst]}]
  
Dilate[lst_, ftr_] :=
  Table[Dilatept[lst[[k]], ftr], {k, 1, Length[lst]}]
  
Ident[lst_] := 
  Table[Identpt[lst[[k]]], {k, 1, Length[lst]}] 
(*
:[font = special1; inactive; preserveAspect; ]
The cell below shows one way to draw a "V".  Evaluate it now.
;[s]
3:0,0;45,1;60,0;62,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,65535,0,0;
:[font = input; preserveAspect; ]
A := {{1,2}, {2,1}, {3,2}}

ShowCurve[A, 4]
:[font = special1; inactive; preserveAspect; ]
The following cell shows how these procedures can be used to manipulate images and display the results.  Evaluate it now.
;[s]
3:0,0;105,1;120,0;122,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,65535,0,0;
:[font = input; preserveAspect; ]
ShowCurves[{A, Mirror[A]}, 4]

ShowCurves[{A, Trans[A, 1, -2]}, 4]

ShowCurves[{A, Rot[A, Pi/2]}, 4]

ShowCurves[{A, Ident[A]}]

ShowCurves[{A, Dilate[A, 2]}]
^*)