(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "NeXT Mathematica Notebook Front End Version 2.2"; NeXTStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e8, 24, "Times"; ; fontset = subtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e6, 18, "Times"; ; fontset = subsubtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, L1, e6, 14, "Times"; ; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L1, a20, 18, "Times"; ; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L1, a15, 14, "Times"; ; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L1, a12, 12, "Times"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-4, 12, "Courier"; ; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-4, 12, "Courier"; ; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R32768, L-4, 12, "Courier"; ; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-4, 12, "Courier"; ; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B32768, L-4, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, noPageBreakInGroup, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, B65535, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = leftheader, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, L1, 12; fontset = leftfooter, 12; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Courier"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; paletteColors = 128; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] Overview of Descartes-Geometry by Denis Monasse Mathsoft Overseas :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Availability :[font = text; inactive; preserveAspect; endGroup] Descartes is a registered trade mark of Mathsoft Overseas, Inc. Copyright Ó 1992, by Mathsoft Overseas, Inc. The program Descartesâ-Geometry is available on any platform at 30 % of the Mathematica price. For information and orders, please contact: Mathsoft Overseas, cp 641, 1211 Geneva 3 (Switzerland). Tel: 41 22 / 346 52 60. Fax: 41 22 / 346 59 39 ;[s] 7:0,0;74,1;76,2;131,3;132,4;186,5;198,6;354,-1; 7:1,16,12,Times,0,18,0,0,0;1,0,0,Symbol,0,18,0,0,0;1,16,12,Times,0,18,0,0,0;1,0,0,Symbol,0,18,0,0,0;1,16,12,Times,0,18,0,0,0;1,16,12,Times,2,18,0,0,0;1,16,12,Times,0,18,0,0,0; :[font = subsection; inactive; preserveAspect; startGroup] Introduction :[font = text; inactive; preserveAspect; endGroup] In the seventeenth century, two french mathematicians, Pierre de Fermat and RenŽ Descartes, realized that algebra and geometry could be unified in a single science that was called analytic geometry. In fact, the recognition for the foundation of the new science went to Descartes although Fermat's work was a little earlier. The reason is due to the formalism used by Descartes who adopted the notation of modern algebra whereas Fermat had kept the cumbersome greek notations. Today, we can resume the treatment of analytic geometry in the context of the modern computers and information languages among which Mathematica is especially well adapted with its outstanding graphics facilities and its functional programming. Let us start :[font = subsection; inactive; preserveAspect; startGroup] Geometric transformations :[font = text; inactive; preserveAspect] The three geometric transformations: translation, rotation and homothety are coded in Descartes-Geometry as operators depending on some arguments. ;[s] 3:0,0;86,1;104,2;147,-1; 3:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = subsubsection; inactive; preserveAspect; startGroup] Translation :[font = text; inactive; preserveAspect] The translation is defined by a vector OM which must have the head POINT in the argument of the operator Shift. In the following example, bt is the point deduced from b by translation defined by AC which is represented by the Descartes primitive VECTOR. ;[s] 17:0,0;39,1;41,2;67,3;72,4;105,5;110,6;138,7;140,8;167,9;168,10;195,11;197,12;226,13;235,14;246,15;252,16;254,-1; 17:1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; startGroup] bt=Shift[c][b] Draw[PointSize[.02],Blue,{a,"A"},{b,"B"},{c,"C"}, VECTOR[a,c], Red,{bt,"B'"}] :[font = output; output; inactive; preserveAspect] POINT[4/3, 2^(1/2)] ;[o] 4 POINT[-, Sqrt[2]] 3 :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 219; pictureHeight = 232] %! %%Creator: Mathematica %%AspectRatio: 1.06066 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.714286 0.0252538 0.714286 [ [(A)] 0.02381 0.02525 -1 -1 Msboxa [(B)] 0.7381 0.02525 -1 -1 Msboxa [(C)] 0.2619 1.03541 -1 -1 Msboxa [(B')] 0.97619 1.03541 -1 -1 Msboxa [ 0 0 0 0 ] [ 1 1.06066 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave grestore gsave 0 0 1 setrgbcolor gsave 0.02 setlinewidth 0.02381 0.02525 Mdot [(A)] 0.02381 0.02525 -1 -1 Mshowa grestore gsave 0.02 setlinewidth 0.7381 0.02525 Mdot [(B)] 0.7381 0.02525 -1 -1 Mshowa grestore gsave 0.02 setlinewidth 0.2619 1.03541 Mdot [(C)] 0.2619 1.03541 -1 -1 Mshowa grestore gsave 0.004 setlinewidth 0.02381 0.02525 moveto 0.2619 1.03541 lineto stroke 0.2619 1.03541 moveto 0.21917 0.98982 lineto 0.25476 1.0051 lineto 0.27978 0.97553 lineto fill grestore 1 0 0 setrgbcolor gsave 0.02 setlinewidth 0.97619 1.03541 Mdot [(B')] 0.97619 1.03541 -1 -1 Mshowa grestore grestore 0 0 moveto 1 0 lineto 1 1.06066 lineto 0 1.06066 lineto closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] The Unformatted text for this cell was not generated. Use options in the Actions Preferences dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = text; inactive; preserveAspect] It must be noted that the geometric transformations may be applied to any type of Graphics object by using a POINT function ;[s] 5:0,0;82,1;91,2;109,3;114,4;124,-1; 5:1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; startGroup] sin=POINT[t,Sin[t]]; sint=Shift[c][sin] ParametricPlot[Evaluate[{{t,Sin[t]},List@@sint}], {t,-Pi,Pi},Axes->False] :[font = output; output; inactive; preserveAspect] POINT[1/3 + t, 2^(1/2) + Sin[t]] ;[o] 1 POINT[- + t, Sqrt[2] + Sin[t]] 3 :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 35; pictureTop = 1; pictureWidth = 296; pictureHeight = 182] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling 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0.43092 lineto Mfstroke grestore grestore grestore % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] The Unformatted text for this cell was not generated. Use options in the Actions Preferences dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = subsubsection; inactive; preserveAspect; startGroup] Rotation :[font = text; inactive; preserveAspect] A rotation of center C and angle t is obtained with the command Rotate[C,t]. In the following example, the circle c2 is rotated about the point A by the angle 2¹/3 and becomes c2'. ;[s] 15:0,0;21,1;22,2;33,3;34,4;64,5;75,6;114,7;116,8;144,9;145,10;159,11;163,12;176,13;179,14;181,-1; 15:1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; startGroup] c2r=Rotate[a,2Pi/3][c2] Draw[Blue,c2,PointSize[.02], {POINT[c2[[1]],c2[[2]]],"c2"},{a,"A"}, Red,c2r,{POINT[c2r[[1]],c2r[[2]]],"c2'"}]; :[font = output; output; inactive; preserveAspect] CIRCLE[-1/4 - (2/3)^(3/2), -2^(3/2)/9 + 3^(1/2)/4, 209^(1/2)/18] ;[o] 3/2 1 2 3/2 -2 Sqrt[3] Sqrt[209] CIRCLE[-(-) - (-) , ----- + -------, ---------] 4 3 9 4 18 :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 289; pictureHeight = 210; endGroup; endGroup] %! %%Creator: Mathematica %%AspectRatio: 0.72953 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.548319 0.328334 0.242086 0.328334 [ [(c2)] 0.71249 0.44846 -1 -1 Msboxa [(A)] 0.54832 0.24209 -1 -1 Msboxa [(c2')] 0.28751 0.28107 -1 -1 Msboxa [ 0 0 0 0 ] [ 1 0.729531 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave grestore gsave 0 0 1 setrgbcolor 0.004 setlinewidth newpath 0.71249 0.44846 0.2637 0 365.73 arc stroke gsave 0.02 setlinewidth 0.71249 0.44846 Mdot [(c2)] 0.71249 0.44846 -1 -1 Mshowa grestore gsave 0.02 setlinewidth 0.54832 0.24209 Mdot [(A)] 0.54832 0.24209 -1 -1 Mshowa grestore 1 0 0 setrgbcolor newpath 0.28751 0.28107 0.2637 0 365.73 arc stroke gsave 0.02 setlinewidth 0.28751 0.28107 Mdot [(c2')] 0.28751 0.28107 -1 -1 Mshowa grestore grestore 0 0 moveto 1 0 lineto 1 0.72953 lineto 0 0.72953 lineto closepath clip newpath % End of Graphics MathPictureEnd :[font = subsubsection; inactive; preserveAspect; startGroup] Homothety :[font = text; inactive; preserveAspect] The homothety given by the operator Zoom needs also two arguments: the center of homothety and the ratio. The triangle tr is transformed by a homothety of center a and ratio 2. ;[s] 9:0,0;36,1;41,2;119,3;121,4;162,5;163,6;173,7;175,8;177,-1; 9:1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; startGroup] trh=Zoom[a,2][tr] Draw[Blue,tr,PointSize[.02],{a,"A"},{b,"B"},{c,"C"}, Red,trh,{trh[[2]],"B'"},{trh[[3]],"C'"}]; :[font = output; output; inactive; preserveAspect] POLYGON[POINT[0, 0], POINT[2, 0], POINT[2/3, 2^(3/2)]] ;[o] 2 3/2 POLYGON[POINT[0, 0], POINT[2, 0], POINT[-, 2 ]] 3 :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 218; pictureHeight = 307; endGroup; endGroup; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1.41421 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.47619 0.0336718 0.47619 [ [(A)] 0.02381 0.03367 -1 -1 Msboxa [(B)] 0.5 0.03367 -1 -1 Msboxa [(C)] 0.18254 0.70711 -1 -1 Msboxa [(B')] 0.97619 0.03367 -1 -1 Msboxa [(C')] 0.34127 1.38054 -1 -1 Msboxa [ 0 0 0 0 ] [ 1 1.41421 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave grestore gsave 0 0 1 setrgbcolor 0.004 setlinewidth 0.02381 0.03367 moveto 0.5 0.03367 lineto 0.18254 0.70711 lineto 0.02381 0.03367 lineto stroke gsave 0.02 setlinewidth 0.02381 0.03367 Mdot [(A)] 0.02381 0.03367 -1 -1 Mshowa grestore gsave 0.02 setlinewidth 0.5 0.03367 Mdot [(B)] 0.5 0.03367 -1 -1 Mshowa grestore gsave 0.02 setlinewidth 0.18254 0.70711 Mdot [(C)] 0.18254 0.70711 -1 -1 Mshowa grestore 1 0 0 setrgbcolor 0.02381 0.03367 moveto 0.97619 0.03367 lineto 0.34127 1.38054 lineto 0.02381 0.03367 lineto stroke gsave 0.02 setlinewidth 0.97619 0.03367 Mdot [(B')] 0.97619 0.03367 -1 -1 Mshowa grestore gsave 0.02 setlinewidth 0.34127 1.38054 Mdot [(C')] 0.34127 1.38054 -1 -1 Mshowa grestore grestore 0 0 moveto 1 0 lineto 1 1.41421 lineto 0 1.41421 lineto closepath clip newpath % End of Graphics MathPictureEnd :[font = section; inactive; preserveAspect; startGroup] The Chinese Star :[font = text; inactive; preserveAspect] Geometry is not limited to abstract constructions, it appears in many human activities such as mechanics, optics, architecture, graphic art, etc. We shall show an animation which describes the generation of a game: the Chinese Star where marbles are gathered on a colored base and must reach the opposite base by jumping over other marbles according to the rule of the symmetry with respect to a point. The game has the shape of a stellated hexagon paved of equilateral triangles. By a set of translations, symmetries and rotations, the full figure can be generated from the central equilateral triangle. ;[s] 3:0,0;219,1;231,2;605,-1; 3:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; startGroup] (* marbles positions and paths *) a1=POINT[0,0]; a2=POINT[1/2,Sqrt[3]/2]; a3=POINT[-1/2,Sqrt[3]/2]; l[0]={POLYGON[a1,a2,a3],{a1,a2,a3}}; l[1]=NestList[Shift[a2],l[0],3]; l[2]=NestList[Drop[#,{Length[#]}]&@ Apply[Shift[a3],#]&, l[1],3]; l3=l[2]/STRAIGHT[0,1,-2Sqrt[3]]; l[3]=Join[l[2],l3]//Flatten//Union; Table[l[i]=Rotate[a1,Pi/3][t[i-1]],{i,4,8}]; Draw[PointSize[.03],#, PlotRange->{{-8,8},{-8,8}}]&/@ FoldList[Union@Flatten@Join[#1,#2]&, l[0], Array[l,8]]; (* bases *) a4=POINT[0,13Sqrt[3]/3];a5=Rotate[a1,Pi/3][a4]; h=RegularPolygon[a4,a5,6,2]; te0=POLYGON[Vertex[h[[1]],1], SideLine[h[[2]],3]*SideLine[h[[1]],1], SideLine[h[[2]],3]*SideLine[h[[1]],3]]; tex=NestList[Rotate[a1,Pi/3],te0,5]; color=Apply[CMYKColor,#]&/@{{1,0,0,0},{0,1,1,0}, {0,1,0,0},{1,0,1,0}, {0,0,1,0},{1,1,0,0}}; Do[tex=Insert[tex,color[[i]],2i-1],{i,6}]; s1=Draw[tex, Fill->True, PlotRange->{{-8,8},{-8,8}}, DisplayFunction->Identity]; s2=Draw[PointSize[.03], NestList[Rotate[a1,Pi/3],t[3],5], PlotRange->{{-8,8},{-8,8}}, DisplayFunction->Identity]; Show[s1,s2,DisplayFunction->$DisplayFunction] ;[s] 5:0,0;1,1;32,2;518,3;527,4;1258,-1; 5:1,10,8,Courier,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Courier,1,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; Cclosed; preserveAspect; pictureLeft = 34; pictureWidth = 284; pictureHeight = 284; startGroup] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0625 0.5 0.0625 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave 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