(*^ ::[ 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; 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; showRuler; 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] Objects :[font = text; inactive; preserveAspect] The basic objects of euclidean geometry are the points, the straight lines, the circles, the open lines and the polygons. To each of those objects correspond a specific head in Descartes-Geometry: POINT, STRAIGHT, CIRCLE, LINE, POLYGON. ;[s] 5:0,0;169,1;173,2;177,3;195,4;237,-1; 5:1,11,8,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,2,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = subsubsection; inactive; preserveAspect; startGroup] POINT :[font = text; inactive; preserveAspect] A point is defined by its two components. :[font = input; preserveAspect; endGroup] a=POINT[0,0]; b=POINT[1,0]; c=POINT[1/3,Sqrt[2]]; :[font = subsubsection; inactive; preserveAspect; startGroup] STRAIGHT :[font = text; inactive; preserveAspect] A straight line is defined by two points (.). ;[s] 3:0,0;42,1;43,2;46,-1; 3:1,11,8,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] d1=a.b d2=a.c :[font = output; output; inactive; preserveAspect] STRAIGHT[0, 1, 0] ;[o] STRAIGHT[0, 1, 0] :[font = output; output; inactive; preserveAspect; endGroup] STRAIGHT[-2^(1/2), 1/3, 0] ;[o] 1 STRAIGHT[-Sqrt[2], -, 0] 3 :[font = text; inactive; preserveAspect] The result is given in the analytic form STRAIGHT[u,v,w] where u, v, w are the coefficients of the straight line equation u x + v y +w = 0 Let us draw the points in blue and the straight lines in red :[font = input; preserveAspect; startGroup] Draw[Blue,PointSize[.02],{a,"A"},{b,"B"},{c,"C"}, Red,d1,d2, Interval->{{-1,2},{-1,2}}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.34127 0.31746 0.34127 0.31746 [ [(A)] 0.34127 0.34127 -1 -1 Msboxa [(B)] 0.65873 0.34127 -1 -1 Msboxa [(C)] 0.44709 0.79023 -1 -1 Msboxa [ 0 0 0 0 ] [ 1 1 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 0.34127 0.34127 Mdot [(A)] 0.34127 0.34127 -1 -1 Mshowa grestore gsave 0.02 setlinewidth 0.65873 0.34127 Mdot [(B)] 0.65873 0.34127 -1 -1 Mshowa grestore gsave 0.02 setlinewidth 0.44709 0.79023 Mdot [(C)] 0.44709 0.79023 -1 -1 Mshowa grestore 1 0 0 setrgbcolor 0.004 setlinewidth 0.02381 0.34127 moveto 0.97619 0.34127 lineto stroke 0.26644 0.02381 moveto 0.49092 0.97619 lineto stroke grestore 0 0 moveto 1 0 lineto 1 1 lineto 0 1 lineto closepath clip newpath % End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect] The points a, b and c are labeled A, B, C respectively and are represented by dots of radius .02 to make them clearly visible. The option Interval defines the frame inside which the objects are plotted in the form {{xmin, xmax},{ymin,ymax}}. You can materialize this frame by adding the option Frame ->True ;[s] 16:0,0;12,1;13,2;15,3;16,4;20,5;22,6;35,7;36,8;38,9;39,10;41,11;42,12;139,13;147,14;296,15;309,-1; 16:1,11,8,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;1,10,8,Times,1,12,0,0,0; :[font = input; preserveAspect; startGroup] Draw[Blue,PointSize[.02],{a,"A"},{b,"B"},{c,"C"}, Red,d1,d2, Interval->{{-1,2},{-1,2}},Frame->True]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; endGroup; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.34127 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0.78571 moveto 1 0.78571 lineto stroke grestore gsave 0.001 setlinewidth 0.99625 0.84921 moveto 1 0.84921 lineto stroke grestore gsave 0.001 setlinewidth 0.99625 0.88095 moveto 1 0.88095 lineto stroke grestore gsave 0.001 setlinewidth 0.99625 0.9127 moveto 1 0.9127 lineto stroke grestore gsave 0.001 setlinewidth 0.99625 0.94444 moveto 1 0.94444 lineto stroke grestore gsave 0.002 setlinewidth 1 0 moveto 1 1 lineto stroke grestore grestore gsave grestore gsave 0 0 1 setrgbcolor gsave 0.02 setlinewidth 0.34127 0.34127 Mdot [(A)] 0.34127 0.34127 -1 -1 Mshowa grestore gsave 0.02 setlinewidth 0.65873 0.34127 Mdot [(B)] 0.65873 0.34127 -1 -1 Mshowa grestore gsave 0.02 setlinewidth 0.44709 0.79023 Mdot [(C)] 0.44709 0.79023 -1 -1 Mshowa grestore 1 0 0 setrgbcolor 0.004 setlinewidth 0.02381 0.34127 moveto 0.97619 0.34127 lineto stroke 0.26644 0.02381 moveto 0.49092 0.97619 lineto stroke grestore 0 0 moveto 1 0 lineto 1 1 lineto 0 1 lineto closepath clip newpath % End of Graphics MathPictureEnd :[font = subsubsection; inactive; preserveAspect; startGroup] CIRCLE :[font = text; inactive; preserveAspect] A circle (<) may be defined by its center and a point ;[s] 3:0,0;10,1;11,2;54,-1; 3:1,11,8,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] c1=aBezier]; Draw[Blue,PointSize[.02],{a,"A"},{b,"B"},{c,"C"}, Red,sp2]; ;[s] 3:0,0;45,1;99,2;105,-1; 3:1,10,8,Times,1,12,0,0,0;1,10,8,Courier,1,12,0,0,0;1,10,8,Times,1,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 211; pictureHeight = 297; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1.41421 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 0.0336718 0.952381 [ [(A)] 0.02381 0.03367 -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 gsave 0.02 setlinewidth 0.02381 0.03367 Mdot [(A)] 0.02381 0.03367 -1 -1 Mshowa grestore 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 1 0 0 setrgbcolor 0.004 setlinewidth 0.02381 0.03367 moveto 0.08981 0.03539 lineto 0.15177 0.04054 lineto 0.20967 0.04913 lineto 0.26352 0.06116 lineto 0.31333 0.07662 lineto 0.35909 0.09552 lineto 0.40079 0.11785 lineto 0.43845 0.14362 lineto 0.47206 0.17283 lineto 0.50162 0.20547 lineto 0.52713 0.24154 lineto 0.54859 0.28106 lineto 0.566 0.324 lineto 0.57937 0.37039 lineto 0.58868 0.42021 lineto 0.59394 0.47347 lineto 0.59516 0.53016 lineto 0.59232 0.59029 lineto 0.58544 0.65385 lineto 0.57451 0.72085 lineto 0.55952 0.79129 lineto 0.54049 0.86516 lineto 0.51741 0.94247 lineto 0.49028 1.02321 lineto 0.4591 1.10739 lineto 0.42387 1.195 lineto 0.3846 1.28605 lineto 0.34127 1.38054 lineto stroke grestore 0 0 moveto 1 0 lineto 1 1.41421 lineto 0 1.41421 lineto closepath clip newpath % End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect; endGroup] It is the option SplineShape which defines the type of spline curve (Cubic is the default). It is prudent to avoid the output of all the points of the spline curves which are usually of no interest by terminating the command with " ; ". ;[s] 7:0,0;17,1;28,2;69,3;74,4;232,5;234,6;237,-1; 7:1,11,8,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 = subsubsection; inactive; preserveAspect; startGroup] POLYGON :[font = text; inactive; preserveAspect] A polygon is a closed line. We define a triangle tr whose vertices are a, b and c denoted A, B and C respectively: ;[s] 15:0,0;49,1;51,2;71,3;72,4;74,5;75,6;80,7;81,8;90,9;91,10;93,11;94,12;98,13;101,14;115,-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] tr=POLYGON[a,b,c] Draw[Blue,PointSize[.02],{a,"A"},{b,"B"},{c,"C"}, Red,tr] :[font = output; output; inactive; preserveAspect] POLYGON[POINT[0, 0], POINT[1, 0], POINT[1/3, 2^(1/2)]] ;[o] 1 POLYGON[POINT[0, 0], POINT[1, 0], POINT[-, Sqrt[2]]] 3 :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 222; pictureHeight = 313] %! %%Creator: Mathematica %%AspectRatio: 1.41421 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 0.0336718 0.952381 [ [(A)] 0.02381 0.03367 -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 gsave 0.02 setlinewidth 0.02381 0.03367 Mdot [(A)] 0.02381 0.03367 -1 -1 Mshowa grestore 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 1 0 0 setrgbcolor 0.004 setlinewidth 0.02381 0.03367 moveto 0.97619 0.03367 lineto 0.34127 1.38054 lineto 0.02381 0.03367 lineto stroke grestore 0 0 moveto 1 0 lineto 1 1.41421 lineto 0 1.41421 lineto closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup; 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- ^*)