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0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 " " 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 5 259 1 {CSTYLE "" -1 -1 "" 1 10 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 5 260 1 {CSTYLE "" -1 -1 "" 1 10 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 5 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 5 262 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 " " 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 1 {PARA 264 "" 0 "" {TEXT -1 6 "Quelle" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Name: Fabian Hust\nSchule: Isolde-Kurz-Gymnasi um\nKlasse: 13\nDatum: 12.11.98\nFach: Mathematik\nThema: Geometrie" } }{PARA 0 "" 0 "" {TEXT -1 57 "Neuere Versionen unter: http://notes.ikg .rt.bw.schule.de/" }}}}{PARA 256 "" 0 "" {TEXT 256 0 "" }}{PARA 258 " " 0 "" {TEXT 256 34 "Geometrische Objekte mit Maple V5\n" }{TEXT 320 37 "September - November 1998\nFabian Hust" }}{PARA 263 "" 0 "" {TEXT 321 18 "*** 1. Fassung ***" }}{PARA 257 "" 0 "" {TEXT -1 339 "\nDieses Referat gibt einen \334berblick zum Arbeiten mit dem geom3d-Package i n Maple V5. Dabei werden die Syntax erl\344utert und wertvolle Tips ge geben. Au\337erdem werden die meisten Beispiele in einem statischen Pl ot und in einer Animation veranschaulicht. F\374r die Experten gibt es au\337erdem jeweils noch einen kurzen Blick \"hinter die Kulissen\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 6 "Punkte" }}{PARA 0 "" 0 "" {TEXT -1 56 "Alles, was man \374ber das A rbeiten mit Punkten wissen mu\337." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 0 "" 0 "" {TEXT 259 30 " Punkte definieren und zeichnen" }}{PARA 0 "" 0 "" {HYPERLNK 17 "2D-Pun kt" 1 "" "2D-Punkt" }{TEXT -1 3 " | " }{HYPERLNK 17 "3D-Punkt" 1 "" "3 D-Punkt" }{TEXT -1 3 " | " }{HYPERLNK 17 "Farbe der Punkte ver\344nder n" 1 "" "Farbe der Punkte ver\344ndern" }{TEXT -1 3 " | " }}{PARA 0 " " 0 "" {TEXT -1 83 "Das Definieren von Punkten erfolgt \374ber den Bef ehl point. Der Syntax hierzu lautet:" }}{PARA 19 "" 0 "" {TEXT -1 63 " point(NameDesPunktes,x-Koordinate, y-Koordinate, z-Koordinate);" }} {PARA 0 "" 0 "" {TEXT -1 78 "F\374r die Erstellung von 2D-Punkten mu \337 man den Namen des Punktes au\337erhalb des " }{TEXT 256 5 "point " }{TEXT -1 93 "-Befehls definieren. Au\337erdem ben\366tigt man f\374 r das Erstellen eines Punktes in 2D das Package " }{TEXT 257 9 "plotto ols" }{TEXT -1 62 ", f\374r das Erstellen eines Punktes im 3D-Raum abe r das Package " }{TEXT 258 6 "geom3d" }{TEXT -1 3 ".\n\n" }{TEXT 261 9 "2D-Punkt\n" }{TEXT 272 126 "\npoint(Name_des_Punktes, x-Wert, y-Wer t);\nWichtig: Die Definition der Punkte im 2D-Bereich mu\337 mit eckig en Klammern erfolgen!\n" }}{EXCHG {PARA 0 "> " 0 "2D-Punkt" {MPLTEXT 1 0 36 "restart:with(plots):with(plottools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "p1:=point([1,2]):\ndisplay(p1);" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 308 33 "Nachgeschaut - So macht das Maple" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "interface(verboseproc=3):\np rint(geom3d[point]);\ninterface(verboseproc=0):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6&%#PPG%\"fG%\"gG%\"hG6#%\"PG6#%inCopyright~(c)~1995~b y~Waterloo~Maple~Inc.~All~rights~reserved.G6\"C'@$309#\"\"#0F2\"\"%-%& ERRORG6#%:wrong~number~of~argumentsG@'333/F2F5-%%typeG6$9%.%*algebraic G-F@6$9&FC-F@6$9'FCC$>&8$6#%.geom3d/coordsG7%FBFGFJ>&FN6#%,geom3d/form G.%(point3dG3/F2F3-F@6$FB7%FCFCFC-%'RETURNG6#-9!6$9$-%#opG6#FB-F76#%8w rong~type~of~argumentsG--%(readlibG6#%0geom3d/isassignG6#F]o-%-setattr ibuteG6$F]o-%%evalG6#FNF]oF-6$FUFPF-" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 9 "\n3D- Punkt" }}{PARA 19 "" 0 "" {TEXT -1 48 "point(Name_des_Punktes, x-Wert, y-Wert, z-Wert);" }}{EXCHG {PARA 0 "> " 0 "3D-Punkt" {MPLTEXT 1 0 33 "restart:with(plots):with(geom3d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "point(p1,0,0,0):\npoint(p2,0,2,2):\npoint(p3,4,4,4): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 19 "" 0 "" {TEXT 286 5 "TIP: " }{TEXT -1 253 "Optionen lassen sich auch als Varia ble abspeichern. So m\374ssen die Einstellungen nur einmal geschrieben werden und nicht bei jedem Plot, wenn alle gleich aussehen sollen. Im folgenden wird dieses Verfahren eingesetzt werden. Hier am Beispiel d er Symbole:" }}{PARA 19 "" 0 "" {TEXT -1 117 "Bezeichnung_der_Option:= symbol=Art_des_Symbols, weitere_Optionen;\nArt_des_Symbols: BOX, CROSS , CIRCLE, POINT, DIAMOND" }}{PARA 19 "" 0 "" {TEXT 306 8 "Hinweis:" } {TEXT -1 191 " Bei 3D-Punkten mu\337 man ein Symbol angeben, damit Map le den Punkt f\374r uns sichtbar darstellt. Gibt man kein Symbol an, s o werden die Punkte so klein, da\337 man sie fast gar nicht erkennen k ann." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 71 "opt:=symbol=circle,axes=frame:\ndisplay(draw (p1),draw(p2),draw(p3),opt);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "Farbe der Punkte ver\344ndern" {TEXT -1 1 "\n " }{TEXT 273 27 "Farbe der Punkte ver\344ndern\n" }{TEXT -1 28 "Leider unterst\374tzt Maple im " }{TEXT 274 4 "draw" }{TEXT -1 140 "-Befehl \+ keine Farben. Deshalb mu\337 man zu einem Trick greifen, um die Farbe \+ von einzelnen Punkten zu ver\344ndern. Man schachtelt einfach einen " }{TEXT 275 7 "display" }{TEXT -1 15 "-Befehl um den " }{TEXT 276 4 "dr aw" }{TEXT -1 38 "-Befehl und kann so die Farbe angeben." }}{PARA 19 " " 0 "" {TEXT -1 217 "display(..., color=Farbe); \nFarbe = aquamarine , black, blue, navy, coral, cyan, brown, gold, green, gray, grey, khak i, magenta, maroon, orange, pink, plum, red, sienna, tan, turquoise, v iolet, wheat, white, yellow " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart:with(plots):with(plottools):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 40 "p1:=point([1,2]):\ndisplay(p1,color=red);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 277 38 "Zufallspunkte erstellen und animieren\n" } {TEXT -1 15 "Mit dem Befehl " }{TEXT 278 9 "randpoint" }{TEXT -1 92 " \+ kann man Punkte per Zufall im Raum plazieren lassen. Diese lassen sich dann auch animieren." }}{PARA 19 "" 0 "" {TEXT -1 67 "randpoint(Name_ des_Punktes.Index, x-Bereich, y-Bereich, z-Bereich);" }}{PARA 0 "" 0 " " {TEXT -1 97 "Im Zusammenspiel mit dem sequence-Befehl lassen sich me hrere Zufalls-Punkte auf einmal erstellen:" }}{PARA 19 "" 0 "" {TEXT -1 48 "seq(randpoint_Befehl, Index=Startwert..Endwert);" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "restart:with(plots):with(geom3d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "Punkteanzahl:=25:\npunkte :=seq(randpoint(pname.i,-10..10,-10..10,-10..10),i=1..Punkteanzahl):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "punkte_plots:=draw(\{punk te\},symbol=circle):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "dis play(seq(display(punkte_plots,orientation=[i,i]),i=-180..180),insequen ce=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 262 33 "Abfragen von Punkte-Eigenschaften" }} {PARA 0 "" 0 "3D-Punkt?" {HYPERLNK 17 "3D-Punkt?" 1 "" "3D-Punkt?" } {TEXT -1 3 " | " }{HYPERLNK 17 "Koordinaten" 1 "" "Koordinaten" } {TEXT -1 3 " | " }{HYPERLNK 17 "Beide Informationen auf einen Blick" 1 "" "Beide Informationen auf einen Blick" }{TEXT 268 12 "\n\n3D-Punkt ?\n" }{TEXT -1 48 "Ob ein Punkt 3D ist, erf\344hrt man mit dem Befehl \+ " }{TEXT 263 4 "form" }{TEXT -1 1 "." }}{PARA 19 "" 0 "" {TEXT -1 23 " form(Name_des_Punktes);" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "r estart:with(geom3d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "poi nt(p1,1,2,3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "form(p1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(point3dG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "Koordinaten" {TEXT 265 11 "Koordinaten" }{TEXT -1 90 "\nDie Koordinaten eines Punktes kann man sich mit dem gl eichnamigen Befehl anzeigen lassen:" }}{PARA 19 "" 0 "" {TEXT -1 30 "c oordinates(Name_des_Punktes);" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "coordinates(p1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"\"\"\" #\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 132 "Die Koordinaten kann man sich aber auch einzeln anzeigen lassen, was insbesondere dann wichtig ist, wenn man mit ihnen rechnen will:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "xcoord(p1);\nycoord(p1);\nzcoord(p1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{PARA 0 "" 0 "Beide Informa tionen auf einen Blick" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 36 " Beide Informationen auf einen Blick\n" }{TEXT -1 15 "Mit dem Befehl " }{TEXT 267 6 "detail" }{TEXT -1 71 " enth\344lt man sowohl die die For m, als auch die Koordinaten des Punktes:" }}{PARA 19 "" 0 "" {TEXT -1 25 "detail(Name_des_Punktes);" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "detail(p1);" }}{PARA 6 "" 1 "" {TEXT -1 25 " name of the objec t: p1" }}{PARA 6 "" 1 "" {TEXT -1 30 " form of the object: point3d" }}{PARA 6 "" 1 "" {TEXT -1 38 " coordinates of the point: [1, 2, 3] " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "Eine weitere M\366glichkeit, Informationen \374ber Punkte abzufragen stell t der Befehl " }{TEXT 305 10 "attributes" }{TEXT -1 5 " dar:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "attributes(p1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7$/%.geom3d/coordsG7%\"\"\"\"\"#\"\" $/%,geom3d/formG%(point3dG" }}}{PARA 0 "" 0 "" {TEXT -1 139 "Maple gib t die Eigenschaften also in der 'table-Form' aus. Wie kommt man aber a n diese Eigenschaften ran? Dies kann uns Maple selbst sagen:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "indices(attributes(p1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$7#%.geom3d/coordsG7#%,geom3d/formG" }} }{PARA 0 "" 0 "" {TEXT -1 80 "Nun wissen wir, wie Maple die Eintr\344g e nennt und k\366nnen diese gezielt ansteuern:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "attributes(p1)[`geom3d/coords`];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"\"\"\"#\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 137 "So haben wir also aus den beiden Informationen die Koordinaten he rausfiltern k\366nnen. Nat\374rlich k\366nnen wir auch die z-Koordinat e auslesen:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "attributes(p1 )[`geom3d/coords`][3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}} {PARA 0 "" 0 "" {TEXT -1 136 "Die in der Liste angegebenen Werte sind \+ durchnummeriert und lassen sich dementsprechend unter Angabe des Liste neintrags wieder auslesen." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 264 31 "Abstand zweier Punkte bes timmen" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Um den Abstand z weier Punkte zu bestimmen, bedient man sich dem Befehl " }{TEXT 269 8 "distance" }{TEXT -1 2 ". " }}{PARA 19 "" 0 "" {TEXT -1 27 "distance(P unkt_1, Punkt_2);" }}{PARA 0 "" 0 "" {TEXT -1 76 "Allgemein berechnet \+ man den Abstand zweier Punkte \"von Hand\" mit der Formel\n" } {XPPEDIT 18 0 "sqrt((xcoord(A)-xcoord(B))^2+(ycoord(A)-ycoord(B))^2+(z coord(A)-zcoord(B))^2);" "6#-%%sqrtG6#,(*$,&-%'xcoordG6#%\"AG\"\"\"-F* 6#%\"BG!\"\"\"\"#F-*$,&-%'ycoordG6#F,F--F66#F0F1\"\"#F-*$,&-%'zcoordG6 #F,F--F>6#F0F1\"\"#F-" }}{PARA 0 "" 0 "" {TEXT -1 78 "Wir definieren u ns zuerst einmal zwei Punkte und berechnen dann deren Abstand:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(geom3d):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "point(p1,2,2,2):\npoint(p2,4 ,4,4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "distance(p1,p2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$-%%sqrtG6#\"\"$\"\"\"\"\"#" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 " " {TEXT 256 14 "Einfacher Plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "with(plots):\no:=symbol=circle:\nsegment(Strecke,[p1,p2]):\ndis play(\ndisplay(draw(p1,o),color=black),\ndisplay(draw(p2,o),color=blac k),\ndisplay(draw(Strecke),color=red)\n);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 256 9 "Animation " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 214 "o:=symbol=circle:\nsegm ent(Strecke,[p1,p2]):\ndisplay(seq(display(\ndisplay(draw(p1,o),color= black),\ndisplay(draw(p2,o),color=black),\ndisplay(draw(Strecke),color =red),\norientation=[i,i]),i=-180..180),insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 256 33 "Nachgeschau t - So macht das Maple" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "in terface(verboseproc=3):\nprint(geom3d[distance]);\ninterface(verbosepr oc=0):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6$%#p1G%#p2G6#%\"fG6#%inCop yright~(c)~1995~by~Waterloo~Maple~Inc.~All~rights~reserved.G6\"C&@$/9# \"\"\"@$-%'memberG6$--%(readlibG6#%,geom3d/formG6#9$<$.%*segment3dG.%+ dsegment3dG-%'RETURNG6#-9!6#-%#opG6#-&%'geom3dG6#.%*DefinedAsGF:@$0F/ \"\"#-%&ERRORG6#%:wrong~number~of~argumentsG>8$-%$mapG6$F6<$F;9%@//FX< #.%(point3dG-%1geom3d/distancepG6$F;Fgn/FX<$F[o.%'line3dG-%3geom3d/dis tancepliGF_o/FX<$.%(plane3dGF[o-%2geom3d/distanceplGF_o/FX<#Fbo-%2geom 3d/distanceskGF_o/FX<$FhoFbo-%2geom3d/distancelpGF_o/FX<#Fho-%2geom3d/ distanceppGF_o-FT6#%8wrong~type~of~argumentsGF+F+F+" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 270 29 "Mitte zweier Punkte bestimmen" }}{PARA 0 "" 0 "" {TEXT -1 120 "Um \+ die Mitte von zwei Punkten zu bestimmen, kann man den Befehl midpoint \+ verwenden. Der Syntax f\374r diesen Befehl lautet:" }}{PARA 19 "" 0 " " {TEXT -1 49 "midpoint(Name_des_Mittelpunktes, Punkt1, Punkt2);" }} {PARA 0 "" 0 "" {TEXT -1 79 "aber auch hier kann man den Mittelpunkt m it einer Formel \"von Hand\" ausrechnen:" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "(ycoord(A)+ycoord(B))/2;" "6#*&,&-%'ycoordG6#%\"AG\"\"\"-F&6#%\" BGF)F)\"\"#!\"\"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(geom3d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "point(p1,2 ,2,2):\npoint(p2,4,4,4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "midpoint(mp,p1,p2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#mpG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 256 33 "Nachgeschaut - So macht das Maple" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 76 "interface(verboseproc=3):\nprint(geom3d[midpoi nt]);\ninterface(verboseproc=0):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6 %%%midpG%\"uG%\"vG6#%$midG6#%inCopyright~(c)~1995~by~Waterloo~Maple~In c.~All~rights~reserved.G6\"C'@$529#\"\"#2\"\"$F1-%&ERRORG6#%:wrong~num ber~of~argumentsG@'3/F1F2/--%(readlibG6#%,geom3d/formG6#9%.%*segment3d G-%'RETURNG6#-9!6$9$-%#opG6#-&%'geom3dG6#.%*DefinedAsGFB33/F1F4/F=.%(p oint3dG/-F>6#9&FZ-&FR6#.%&pointG6$.8$-%$zipG6%R6$%\"xG%\"yGF,6$%)opera torG%&arrowGF,,&FL#\"\"\"F2FCF]pF,F,F,--F?6#%3geom3d/coordinatesGFB-F` pFhn-F66#%8wrong~type~of~argumentsG--F?6#%0geom3d/isassignG6#FL-%-seta ttributeG6$FL-%+attributesG6#FaoFLF,F,F," }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 144 "Man erh\344lt zun \344chst kein Ergebnis, da Maple den Mittekpunkt in einer Variablen al s Punktekoordinaten abspeichert. Diese kann man mit dem Befehl " } {TEXT 271 11 "coordinates" }{TEXT -1 10 " anzeigen." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "coordinates(mp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 27 "Mit dem bo olschen Operator " }{TEXT 307 2 "is" }{TEXT -1 201 " kann man einfach \+ \374berpr\374fen, ob der Punkt mp wirklich der Mittelpunk der beiden P unkte p1 und p2 ist. Dies \374berpr\374ft man, indem man schaut, ob de r Abstand p1 zu mp gleich dem Abstand von p2 zu mp ist." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "is(distance(p1,mp)=distance(p2,mp)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 40 "Dies kann man auch graphisch darstellen:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 282 14 "Einfacher Plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "o:=symbol=circle:\ndisplay(\ndisplay(draw(p1,o),color=black),\ndi splay(draw(p2,o),color=black),\ndisplay(draw(mp,o),color=red)\n);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT 283 9 "Animation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "o:=symbol=circle:\ndisplay(seq(display(\ndisplay(draw(p1,o),color=bla ck),\ndisplay(draw(p2,o),color=black),\ndisplay(draw(mp,o),color=red), \norientation=[i,i]),i=-180..180),insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 279 31 "Zentrum von n Punkten bestimmen" }}{PARA 19 "" 0 "" {TEXT -1 62 "centroid(Name_des_Punktes, Liste_der_zu_\374berpr\374fenden-Punkte);" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 171 "restart:with(geom3d):\nPunkte := [point(A,0,0,0),p oint(B,1,0,0),point(C,0,1,0),point(D,0,0,1)]:\ngtetrahedron(ABCD, Punk te):\ncentroid(Zentrum, Punkte):\ncoordinates(Zentrum);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7%#\"\"\"\"\"%F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 96 "Dies kann man nat\374rlich auch wieder plotten, um das Ergebnis gr aphisch nachvollziehen zu k\366nnen:" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 280 14 "Einfacher Plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 197 "w ith(plots):\npoint(z,1/4,1/4,1/4):\no:=symbol=circle,color=black:\ndis play(\ndisplay(draw(A),o),\ndisplay(draw(B),o),\ndisplay(draw(C),o),\n display(draw(D),o),\ndisplay(draw(z),symbol=circle,color=red)\n);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT 281 9 "Animation" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 257 "with(plots):\npoint(z,1/4,1/4,1/4):\no:=symbol=circl e,color=black:\ndisplay(seq(display(\ndisplay(draw(A),o),\ndisplay(dra w(B),o),\ndisplay(draw(C),o),\ndisplay(draw(D),o),\ndisplay(draw(z),sy mbol=circle,color=red),\norientation=[i,i]),i=-180..180),insequence=tr ue);" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 125 "Nat\374rlich kann man sich auch Zufallspunkte erstelle n lassen und dann von diesen den Mittelpunkt (=Schwerpunkt) suchen las sen:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "seq(randpoint(p.i,-1 0..10,-10..10,-10..10),i=1..30):\ncentroid(Zentrum,[seq(p.i,i=1..30)]) :" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 256 14 "Einfacher Plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "with(plots):\ndisplay(\ndisplay(\n draw(\{seq(p.i,i=1..30)\},symbol=diamond),color=blue),\ndisplay(draw(Z entrum,symbol=diamond),color=red),axes=boxed,view=[-10..10,-10..10,-10 ..10]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 256 9 "Animation" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "with(plots):\npoint(z,1/4,1/4,1/4):\no:= symbol=circle,color=black:\ndisplay(seq(display(\ndisplay(\ndraw(\{seq (p.i,i=1..30)\},symbol=diamond),color=blue),\ndisplay(draw(Zentrum,sym bol=diamond),color=red),\norientation=[i,i]),i=-180..180),insequence=t rue);" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 256 33 "Nachgeschaut - So macht das Maple" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "interface(verboseproc=3):\np rint(geom3d[centroid]);\ninterface(verboseproc=0):" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#R6$%\"cG%\"vG6)%\"fG%%versG%\"nG%\"XG%\"YG%\"ZG%\"iG6 #%inCopyright~(c)~1995~by~Waterloo~Maple~Inc.~All~rights~reserved.G6\" C%@$09#\"\"#-%&ERRORG6#%:wrong~number~of~argumentsG>8$-&%'geom3dG6#.%% formG6#9%@'5/F<.%/gtetrahedron3dG/F<.%+triangle3dGC(>8%-&F?6#.%)vertic esGFC>8&-%%nopsG6#FO>8'*&-%$addG6$8*/Fjn-%$mapG6$R6#%\"xGF16$%)operato rG%&arrowGF1-%#opG6$\"\"\"9$F1F1F1FO\"\"\"FV!\"\">8(*&-Fhn6$Fjn/Fjn-F] o6$RF`oF1FboF1-Ffo6$F6FioF1F1F1FOFjoFVF[p>8)*&-Fhn6$Fjn/Fjn-F]o6$RF`oF 1FboF1-Ffo6$\"\"$FioF1F1F1FOFjoFVF[p-%'RETURNG6#-&F?6#.%&pointG6&FioFe nF]pFhp3-%%typeG6$FD.%%listG0-FXFC\"\"!C$>F<-F]o6$F>-%(convertG6$FD.%$ setG@%/F<<#.%(point3dGC'>FVFcr>Fen*&-Fhn6$Fjn/Fjn-F]o6$&F?6#.%'xcoordG FDFjoFVF[p>F]p*&-Fhn6$Fjn/Fjn-F]o6$&F?6#.%'ycoordGFDFjoFVF[p>Fhp*&-Fhn 6$Fjn/Fjn-F]o6$&F?6#.%'zcoordGFDFjoFVF[pFcq-F86#%Delements~of~the~list ~must~be~pointsG-F86#%8wrong~type~of~argumentsGF1F1F1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 284 47 "Untersuchen, ob Punkte auf einer Geraden liegen" }}{PARA 0 "" 0 "" {TEXT -1 93 "Es gilt zu untersuchen, ob sich Punkte auf einer Ger aden befinden. Hierzu gibt es den Befehl " }{TEXT 285 11 "IsOnObject \+ " }{TEXT -1 2 "im" }{TEXT 287 15 " geom3d-package" }}{PARA 19 "" 0 "" {TEXT -1 73 "IsOnObject(zu_\374berpr\374fender_Punkt, Gerade_auf_der_d er_Punkt_liegen_soll);" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "re start:with(geom3d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "poin t(p1,0,0,0):\npoint(p2,0,2,2):\npoint(p3,4,4,4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "line(l2,[p1,p2]):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 56 "IsOnObject(p1,l2);\nIsOnObject(p2,l2);\nIsOnObject( p3,l2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 84 "Diesen Sachverhalt kann man jetzt auch wieder im Plot ans chaulich darstellen lassen:" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 288 14 "E infacher Plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "with(plots) :\nopt:=symbol=circle:\ndisplay(draw(p1),draw(p2),draw(p3),draw(l2),op t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 289 9 "Animation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "with(plots):\nopt:=symbol=circle:\ndisplay(seq(display(\ndisp lay(draw(p1),opt),\ndisplay(draw(p2),opt),\ndisplay(draw(p3),opt),\ndi splay(draw(l2)),\norientation=[i,i]),i=-180..180),insequence=true);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 256 33 "Nachge schaut - So macht das Maple" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "interface(verboseproc=3):\nprint(geom3d[IsOnObject]);\ninterface(v erboseproc=0):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6%%\"uG%\"vG'%%cond G%%nameG6\"6#%inCopyright~(c)~1995~by~Waterloo~Maple~Inc.~All~rights~r eserved.GF*C%@$529#\"\"#2\"\"$F1-%&ERRORG6#%:wrong~number~of~arguments G@$54-%'memberG6$--%(readlibG6#%,geom3d/formG6#9%<%.%'line3dG.%)sphere 3dG.%(plane3dG30-F@6#9$.%(point3dG3-%%typeG6$FQ<$.%%listG.%$setG0-%$ma pG6$F@<#-%#opGFP<#FR-F66#%8wrong~type~of~argumentsG@%/F?FG-%.geom3d/on objlG6#9\"-%/geom3d/onobjpsGFfoF*F*F*" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 290 44 "Untersuche n, ob Punkte in einer Ebene liegen" }}{PARA 0 "" 0 "" {TEXT -1 88 "Es \+ gilt zu untersuchen, ob Punkte auf einer Ebene liegen. Auch dies geht \+ mit dem Befehl " }{TEXT 256 10 "IsOnObject" }{TEXT 291 2 ":\n" }{TEXT 292 8 "Hinweis:" }{TEXT 293 1 " " }{TEXT 257 11 "Siehe auch " } {HYPERLNK 17 "hier" 1 "" "Ebene" }{TEXT 294 60 " nach, um \374ber die \+ Konstruktion von Ebenen mehr zu erfahren!" }}{PARA 19 "" 0 "" {TEXT -1 72 "IsOnObject(zu_\374berpr\374fender_Punkt, Ebene_auf_der_der_Punk t_liegen_soll);" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:w ith(geom3d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "point(p1,0, 0,0):\npoint(p2,0,2,2):\npoint(p3,4,4,4):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "plane(ebene,[p1,p2,p3]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "IsOnObject(p1,ebene);\nIsOnObject(p2,ebene);\nIsOn Object(p3,ebene);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% %trueG" }}}{PARA 0 "" 0 "" {TEXT -1 21 "Darstellung als Plot:" }} {SECT 1 {PARA 5 "" 0 "" {TEXT 256 14 "Einfacher Plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "with(plots):\nopt:=symbol=circle,color=b lue:\ndisplay(display(draw(p1),opt),display(draw(p2),opt),display(draw (p3),opt),display(draw(ebene),style=patchnogrid,color=coral),orientati on=[-11,101]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT 256 9 "Animation" }{TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 233 "with(plots):\nopt:=symbol=d iamond,color=blue:\ndisplay(seq(display(\ndisplay(draw(p1),opt),\ndisp lay(draw(p2),opt),\ndisplay(draw(p3),opt),\ndisplay(draw(ebene),style= patchnogrid,color=coral),\norientation=[i,i]),i=20..70),insequence=tru e);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 "Strecke n" }}{PARA 0 "" 0 "" {TEXT -1 57 "Alles, was man \374ber das Arbeiten \+ mit Strecken wissen mu\337." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 295 17 "Strecken zeichnen" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 37 "Eine Strecke kann man mit dem Befehl " } {TEXT 296 7 "segment" }{TEXT -1 31 " zeichnen, das sich im Package " } {TEXT 297 6 "geom3d" }{TEXT -1 10 " befindet." }}{PARA 19 "" 0 "" {TEXT -1 46 "segment(Name_der_Strecke, [Punkt_1, Punkt_2]):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(geom3d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "point(p1,0,0,0):\npoint(p2,0,2,2): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "segment(Strecke,[p1,p2] ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 " " 0 "" {TEXT 256 14 "Einfacher Plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "with(plots):\no:=symbol=circle:\ndisplay(\ndisplay(d raw(p1,o),color=black),\ndisplay(draw(p2,o),color=black),\ndisplay(dra w(Strecke),color=red)\n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 256 9 "Animation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 203 "with(plots):\no:=symbol=circle:\ndisplay (seq(display(\ndisplay(draw(p1,o),color=black),\ndisplay(draw(p2,o),co lor=black),\ndisplay(draw(Strecke,o),color=red),\norientation=[i,i]),i =-180..180),insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 256 33 "Nachgeschaut - So macht das Maple" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "interface(verboseproc=3):\np rint(geom3d[segment]);\ninterface(verboseproc=0):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6%%%usegG%#P1G%#P2G6#%$segG6#%inCopyright~(c)~1995~by~ Waterloo~Maple~Inc.~All~rights~reserved.G6\"C(@$329#\"\"#2\"\"$F1-%&ER RORG6#%:wrong~number~of~argumentsG@%/F1F2@$54-%%typeG6$&9\"6#F2.%%list G0-%$mapG6$-%(readlibG6#%,geom3d/formGFA7$.%(point3dGFO-F66#%8wrong~ty pe~of~argumentsG@$50-FJ6#FAFO0-FJ6#&FB6#F4FOFQ@'/F1F4C$>&8$FL.%*segmen t3dG>&F]o6#%-geom3d/givenG7$9%9&F:-%'RETURNG6#-9!6$9$-%#opG6#FeoFQ--FK 6#%0geom3d/isassignG6#F]p-%-setattributeG6$F]p-%%evalG6#F]oF]pF,6$FMFc oF," }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 298 33 "Parallelen zu den Achsen zeichnen" }}{PARA 0 " " 0 "" {TEXT -1 95 "Um die Koordinaten-Achsen zu veranschaulichen, sol len Parallelen zu den Achsen erstellt werden." }}{SECT 1 {PARA 259 "" 0 "" {TEXT -1 14 "Einfacher Plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(geom3d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "point(A,0.5,0.5,0):\npoint(B,4,0.5,0):\npoint(C,0.5,4,0):\npoint (D,0.5,0.5,4):\npoint(E,0,0,0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "segment(AD,[A,D]):\nsegment(AC,[A,C]):\nsegment(AB,[A,B]):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "dr1:=display(draw(\{A,B,C, D,E\},symbol=circle,scaling=constrained),color=blue):\ndr2:=display(dr aw(\{AD,AC,AB\}),color=red):\nwith(plots):\ndisplay(dr1,dr2,axes=norma l,orientation=[14,49]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 9 "Animation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(geom3d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "point(A,0.5,0.5,0):\npoint(B,4,0.5,0):\npoi nt(C,0.5,4,0):\npoint(D,0.5,0.5,4):\npoint(E,0,0,0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "segment(AD,[A,D]):\nsegment(AC,[A,C]):\ns egment(AB,[A,B]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "dr1:= display(draw(\{A,B,C,D,E\},symbol=circle,scaling=constrained),color=bl ue):\ndr2:=display(draw(\{AD,AC,AB\}),color=red):\nwith(plots):\ndispl ay(seq(display(dr1,dr2,axes=normal,orientation=[i,i]),i=-180..180),ins equence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT 299 29 "L\344nge einer Strecke bestimmen" }}{PARA 0 "" 0 "" {TEXT -1 49 "Die L\344nge einer Strecke l\344\337t sich mit dem Befehl " }{TEXT 300 8 "distance" }{TEXT -1 79 " bestimmen. Dabei kann man zw ei Punkte, bzw. eine definierte Strecke einsetzen." }}{PARA 19 "" 0 " " {TEXT -1 27 "distance(Name_der_Strecke);" }}{PARA 0 "" 0 "" {TEXT -1 4 "oder" }}{PARA 19 "" 0 "" {TEXT -1 27 "distance(Punkt_1, Punkt_2) ;" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(geom3d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "point(p1,0,0,0):\npoint(p 2,0,2,2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "segment(Strecke,[p1,p2]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "distance(Strecke);\ndistance (p1,p2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$-%%sqrtG6#\"\"#\"\"\"F (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$-%%sqrtG6#\"\"#\"\"\"F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 256 33 "Nachgeschaut - So macht das Maple" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 76 "interface(verboseproc=3):\nprint(geom3d[distan ce]);\ninterface(verboseproc=0):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6 $%#p1G%#p2G6#%\"fG6#%inCopyright~(c)~1995~by~Waterloo~Maple~Inc.~All~r ights~reserved.G6\"C&@$/9#\"\"\"@$-%'memberG6$--%(readlibG6#%,geom3d/f ormG6#9$<$.%*segment3dG.%+dsegment3dG-%'RETURNG6#-9!6#-%#opG6#-&%'geom 3dG6#.%*DefinedAsGF:@$0F/\"\"#-%&ERRORG6#%:wrong~number~of~argumentsG> 8$-%$mapG6$F6<$F;9%@//FX<#.%(point3dG-%1geom3d/distancepG6$F;Fgn/FX<$F [o.%'line3dG-%3geom3d/distancepliGF_o/FX<$F[o.%(plane3dG-%2geom3d/dist anceplGF_o/FX<#Fbo-%2geom3d/distanceskGF_o/FX<$FboFho-%2geom3d/distanc elpGF_o/FX<#Fho-%2geom3d/distanceppGF_o-FT6#%8wrong~type~of~argumentsG F+F+F+" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 301 33 "Teilpunkte von Strecken bestimmen" }} {PARA 0 "" 0 "" {TEXT -1 69 "Um Teilpunkte einer Strecke zu erhalten, \+ bedient man sich dem Befehl " }{TEXT 302 9 "OnSegment" }{TEXT -1 108 " . Man erh\344lt als Ergebnis einen Punkt, der mit dem ersten Punkt zus ammen den x-ten Teil der Strecke bildet. " }{TEXT 303 10 "OnSegment " }{TEXT -1 202 "teilt die Strecke im Verh\344ltnis x:1. Will man also d ie Strecke halbieren, so gibt man 1 an, will man die Strecke dritteln, dann gibt man 2 ein. Beim Halbieren haben die Strecken also das Verh \344ltnis 1:1." }}{PARA 19 "" 0 "" {TEXT -1 66 "OnSegment(Name_des_Pun ktes, Name_der_Strecke, Teilungsverh\344ltnis):" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " restart:with(geom3d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "po int(p1,0,0,0):\npoint(p2,2,2,2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "segment(Strecke,[p1,p2]):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 27 "OnSegment(wert,Strecke, 1):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "punkt:=coordinates(wert);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&punktG7%\"\"\"F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 14 "Einfacher \+ Plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "p3:=display(draw(point(p3,co ordinates(wert)),symbol=circle),color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "dr1:=display(draw(Strecke),color=red):\ndr2:=di splay(draw(\{p1,p2\},symbol=circle),color=red):\ndisplay(dr1,dr2,p3,or ientation=[-45,90]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 9 "Animation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "p3:=display(draw(point(p3,coordinates(wert)),symbol=c ircle),color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "dr1: =display(draw(Strecke),color=red):\ndr2:=display(draw(\{p1,p2\},symbol =circle),color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "dis play(seq(display(dr1,dr2,p3,axes=boxed,orientation=[i,i]),i=-180..180) ,insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 256 33 "Nachgeschaut - So macht das Maple" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 77 "interface(verboseproc=3):\nprint(geom3d[OnSegm ent]);\ninterface(verboseproc=0):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R 6&%#qqG%\"uG%\"vG%\"kG6(%\"qG%$msgG%#skG%#cuG%#cvG%\"iG6#%inCopyright~ (c)~1995~by~Waterloo~Maple~Inc.~All~rights~reserved.G6\"C'@$529#\"\"$2 \"\"%F7-%&ERRORG6#%:wrong~number~of~argumentsG@'333/F7F:/--%(readlibG6 #%,geom3d/formG6#9%.%(point3dG/-FF6#9&FL-%%typeG6$9'.%*algebraicGC'>8& --FG6#%1geom3d/equalzeroG6#,&FU\"\"\"F[oF[o@&/FZ%%trueGC$>8%-%(sprintf G6$%Gthe~ratio~%a~must~be~different~from~-1GFU-F<6#Fao/FZ%%FAILGC$>Fao -Fco6$%Bunable~to~determine~if~%a~is~zeroGFjnFfo>8'--FG6#%3geom3d/coor dinatesGFJ>8(-FbpFP-&%'geom3dG6#.%&pointG6$.8$-%$seqG6$*&,&&F`p6#8)F[o *&FUF[o&FfpFgqF[oF[o\"\"\"Fjn!\"\"/Fhq;F[oF833/F7F8/FE.%*segment3dG-FS 6$FQFV-%'RETURNG6#-9!6%9$-%#opG6#-&Fjp6#.%*DefinedAsGFJFQ-F<6#%8wrong~ type~of~argumentsG--FG6#%0geom3d/isassignG6#F]s-%-setattributeG6$F]s-% +attributesG6#F`qF]sF2F2F2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 304 25 "Schar paralleler Streck en" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart:with(plots):wi th(plottools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Anzahl:=5 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for i from 1 to Anzah l do\n pfeil[i]:=arrow([i,0], [i,10], .2, .9, .1, color=blue):\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "display(seq(display(pfeil [i]),i=1..Anzahl));" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 309 96 "Berechn e die Winkel, die Vektoren mit den drei Koordinatenachsen einschlie \337en (Richtungskosinus)" }}{PARA 0 "" 0 "" {TEXT -1 139 "Hierzu defi nieren wir uns eine Strecke im 3D-Raum. Maple nimmt uns viel Denk- und Rechenarbeit ab, da es selbst f\374r diesen Fall im Package " }{TEXT 310 6 "geom3d" }{TEXT -1 163 " einen Befehl gibt. Hierf\374r m\374ssen wir aber noch einiges an Vorarbeit leisten. Sowohl die Strecke, als a uch die Koordinatenachsen m\374ssen als Gerade definiert sein!" }} {PARA 19 "" 0 "" {TEXT -1 57 "FindAngle(Name_der_Geraden_Nr_1, Name_de r_Geraden_Nr_2); " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "restart :with(geom3d):\npoint(p1,2,6,1):\npoint(p2,3,5,2):\nsegment(Strecke,[p 1,p2]):" }}}{PARA 0 "" 0 "" {TEXT -1 43 "Nun machen wir aus der Streck e eine Gerade:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "line(Gerad e,[DefinedAs(Strecke)[1],DefinedAs(Strecke)[2]]):" }}}{PARA 0 "" 0 "" {TEXT -1 48 "Und stellen das ganze erst einmal graphisch dar:" }} {SECT 1 {PARA 256 "" 0 "" {TEXT -1 14 "Einfacher Plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 130 "dr1:=display(draw(Strecke),color=red):\ndr2:=displ ay(draw(\{p1,p2\},symbol=circle),color=red):\ndisplay(dr1,dr2,orientat ion=[-45,90]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 9 " Animation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "dr1:=display(draw(Strecke) ,color=red):\ndr2:=display(draw(\{p1,p2\},symbol=circle),color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "display(seq(display(dr1,d r2,axes=none,orientation=[i,i]),i=-180..180),insequence=true);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 38 "Da ja wie oben bes chrieben der Befehl " }{TEXT 256 9 "FindAngle" }{TEXT -1 100 " nur mit Geraden zusammenarbeitet, m\374ssen wir auch noch die Koordinatenachs en in Geraden darstellen." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "point(gw1,0,0,0):\npoint(gw2,0,0,1):\npoint(gw3,0,1,0):\npoint(gw 4,1,0,0):\nline(xg,[gw1,gw2]):\nline(yg,[gw1,gw3]):\nline(zg,[gw1,gw4] ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "`Die Winkel haben de n folgenden Betrag:`;\n`\257\257\257\257\257\257\257\257\257\257\257 \257\257\257\257\257\257\257\257\257\257\257\257\257\257\257\257\257 \257\257\257\257`;\n`\267 x-Achse: `,evalf(FindAngle(Gerade,xg));\n` \267 y-Achse: `,evalf(FindAngle(Gerade,yg));\n`\267 z-Achse: `,evalf(F indAngle(Gerade,zg));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%GDie~Winkel~ haben~den~folgenden~Betrag:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%A|ju| ju|ju|ju|ju|ju|ju|ju|ju|ju|ju|ju|ju|ju|ju|ju|ju|ju|ju|ju|ju|ju|ju|ju|j u|ju|ju|ju|ju|ju|ju|juG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,|bv~x-Ach se:~G$\"+!=mJb*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,|bv~y-Achse:~G $\"+!=mJb*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,|bv~z-Achse:~G$\"+! =mJb*!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "Geraden" }}{PARA 0 "" 0 "" {TEXT -1 55 "Alles was man \374ber das Arbeiten mit Geraden wissen mu \337." }}{SECT 1 {PARA 4 "" 0 "" {TEXT 311 102 "Definiere Geraden auf \+ verschieden Arten und zeichne sie. Zeichne jeweils die definierenden O bjekte mit" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 312 24 "Gerade durch zwei \+ Punkte" }}{PARA 0 "" 0 "" {TEXT -1 115 "Um eine Gerade durch zwei Punk te definieren zu k\366nnen, m\374ssen wir uns nat\374rlich erst einmal zwei Punkte definieren:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " restart:with(geom3d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "po int(p1,1,2,3):\npoint(p2,4,4,4):" }}}{PARA 0 "" 0 "" {TEXT -1 46 "Dies e Punkte verbinden wir nun mit dem Befehl " }{TEXT 313 4 "line" } {TEXT -1 29 " und erhalten so eine Gerade:" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "line(Gerade,[p1,p2]):" }}}{SECT 1 {PARA 256 "" 0 " " {TEXT -1 14 "Einfacher Plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "dr1: =display(draw(Gerade),color=red):\ndr2:=display(draw(\{p1,p2\},symbol= circle),color=red):\ndisplay(dr1,dr2,orientation=[-45,90]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 9 "Animation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "dr1:=displa y(draw(Gerade),color=red):\ndr2:=display(draw(\{p1,p2\},symbol=circle) ,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "display(seq (display(dr1,dr2,axes=none,orientation=[i,i]),i=-180..180),insequence= true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 256 33 "Nachgeschaut - So macht das Maple" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "interface(verboseproc=3):\nprint(geom3d[line]);\ninte rface(verboseproc=0):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6%%#llG%\"MG '%\"nG%%nameG66%\"lG%$disG%$msgG%#n3G%\"xG%\"yG%\"zG%\"tG%#x0G%#x1G%#y 0G%#y1G%#z0G%#z1G%#d1G%#d2G%#d3G%$tmpG%#f1G%#f2G6#%inCopyright~(c)~199 5~by~Waterloo~Maple~Inc.~All~rights~reserved.G6\"C)@$529#\"\"#2\"\"$FF -%&ERRORG6#%:wrong~number~of~argumentsG@$33-%%typeG6$&9\"6#FG.%%listG0 -%%nopsG6#FTFG0FZFI-FK6#%8wrong~type~of~argumentsG@%/FFFIC$>&8$6#%/geo m3d/indnameG9&>8'%%trueG>Feo%&falseG@&/FZFGC%>86--%(readlibG6#%,geom3d /formG6#&9%6#\"\"\">87-F_p6#&FepFV@33/F]p.%(point3dG/FipF`qC)>8&-%(spr intfG6#%@define~the~line~from~two~pointsG-%)userinfoG6%FG%'geom3dGFeq> 8%--F`p6#%0geom3d/distinctG6#Fep@&/F_rFhoC$>Feq-Fgq6$%Lthe~two~given~p oints~%a~and~%a~are~the~sameG-%#opGFdr-FK6#Feq/F_r%%FAILGC$>Feq-Fgq6$% Nunable~to~determine~if~%a~and~%a~are~distinctGF\\sF^s>&F`oFap.%'line3 dG>&F`o6#%.geom3d/fixedpGFdp>&F`o6#%-geom3d/paravG--F`p6#%2geom3d/vect orformG6#--F`p6#%0geom3d/dsegmentG6$.85F\\s3F_q5-FR6$F\\q.%'vectorG-FR 6$F\\qFWC&>Feq-Fgq6#%_pdefine~the~line~which~passes~a~fixed~point~and~ ~~~~~~~~~~~~~~~~~parallel~to~a~vectorGFjq>F_r--F`p6#%2geom3d/iszerovec tGF[q@'/F_rFfoC$>Feq-Fgq6$%=%a~must~be~a~non-zero~vectorGF\\qF^sF`sC$> Feq-Fgq6$%Kunable~to~determine~if~%a~is~a~zero~vectorGF\\qF^sC%>FhsFis >F\\tFdp>F`t-%(convertGFfu35-FR6$FdpFcu-FR6$FdpFWFbq@%Feo-%'RETURNG6#- 9!6%&FUFfp7$F\\qFdp&FU6#FI-F[x6#-F^x6$F`xFax3F_q/Fip.%+dsegment3dG@%Fe o-F[x6#-F^x6%F`x7$Fdp-FdtF[qFbx-F[x6#-F^x6$F`xFay3/F]pFjxFbq@%FeoFjwFd x3F_q/Fip.%(plane3dGC(-F^x6%F`x7$Fdp-&F]r6#.%-NormalVectorGF[q&FU6#;FI FF>8(--F`p6#%-geom3d/xnameGF[q>8)--F`p6#%-geom3d/ynameGF[q>8*--F`p6#%- geom3d/znameGF[q@$330F[[lFas0Fa[lFas0Fg[lFasC%>&-%+attributesG6#F`xF^[ lF[[l>&Fe\\lFd[lFa[l>&Fe\\lFj[lFg[l-F[xFg\\l3/F]pF\\zFbqC(-F^x6%F`xFax Fgz>F[[lF\\[l>Fa[lFb[l>Fg[lFh[l@$F]\\lC%>Fd\\lF[[l>Fi\\lFa[l>F[]lFg[lF \\]l3F^]lF[zC'-&F]r6#.%-intersectionG6$F`oF\\s@$0-F_p6#F`oFis-FK6#%Bth e~two~given~planes~are~parallelG--F`p6#%0geom3d/isassignG6#9$-%-setatt ributeG6$F^_l-Ff\\lFe^l-F[xF]_lFhn/FZFIC*@%F\\o>8+FbxC$>Fh_l--F`p6#%/g eom3d/getnameG6$Fis.Fh_l>F_oFh_l>F[[l-F]s6$FgpFep>Fa[l-F]s6$FGFep>Fg[l -F]s6$FIFep@%/F[[l\"\"!>82F]al>F_al-%'degreeG6$F[[lFh_l@%/Fa[lF]al>83F ]al>Fgal-Fbal6$Fa[lFh_l@%/Fg[lF]al>84F]al>F^bl-Fbal6$Fg[lFh_l@%333-%'m emberG6$F_al<$F]alFgp-Fgbl6$FgalFibl-Fgbl6$F^blFibl0<%F_alFgalF^bl<#F] alC+>8,-%&coeffG6%F[[lFh_lF]al>8--Fecl6%F[[lFh_lFgp>8.-Fecl6%Fa[lFh_lF ]al>8/-Fecl6%Fa[lFh_lFgp>80-Fecl6%Fg[lFh_lF]al>81-Fecl6%Fg[lFh_lFgp>Fh sFis>F\\t-&F]r6#.%&pointG6$--F`p6#%-geom3d/gnameG6#-%$catG6$.%'fixed_G F^_l7%FcclF\\dlFddl>F`t7%FhclF`dlFhdlFhnFi^l-F`_l6$F^_l-%%evalGFe^lF^_ lFA6&%-geom3d/hnameG%-geom3d/vnameGFbp%.geom3d/coeffsGFA" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 257 41 "Gerade durch einen Punkt und einen Vektor" }}{PARA 0 "" 0 "" {TEXT -1 238 "Um eine Gerade durch einen Punkt und einen Vektor defini eren zu k\366nnen, m\374ssen wir uns nat\374rlich erst einmal einen Pu nkt und einen Vektor definieren. Einen Vektor kann man mit dem gleichn amigen Befehl definieren. Die Syntax hierzu lautet:" }}{PARA 19 "" 0 " " {TEXT -1 49 "Name_des_Vektors:=vector([x-Wert,y-Wert,z-Wert]);" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(geom3d):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "point(p1,1,2,3):\nv1:=vector ([4,4,4]):" }}}{PARA 0 "" 0 "" {TEXT -1 46 "Diese Punkte verbinden wir nun mit dem Befehl " }{TEXT 258 4 "line" }{TEXT -1 29 " und erhalten \+ so eine Gerade:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "line(Gera de,[p1,v1]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 14 "Einfacher Plot" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "dr1:=display(draw(Gerade),color=red):\ndr2:=display( draw(\{p1\},symbol=circle),color=red):\ndisplay(dr1,dr2,orientation=[- 45,90]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 9 "Animation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "dr1:=display(draw(Gerade),color=red):\ndr2:=display(draw(\{p1\}, symbol=circle),color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "display(seq(display(dr1,dr2,axes=none,orientation=[i,i]),i=-180..1 80),insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 261 "" 0 "" {TEXT 256 52 "Ger ade durch einen Punkt und ein gerichtetes Segment" }}{PARA 0 "" 0 "" {TEXT -1 209 "Um eine Gerade durch einen Punkt und ein gerichtetes Seg ment definieren zu k\366nnen, m\374ssen wir uns nat\374rlich erst einm al einen Punkt und ein gerichtetes Segment definieren. Einen Vektor ka nn man mit dem Befehl " }{TEXT 314 8 "dsegment" }{TEXT -1 38 " definie ren. Die Syntax hierzu lautet:" }}{PARA 19 "" 0 "" {TEXT -1 75 "Name_d es_gerichteten_Segments:=dsegment(Name_des_Segments,[Punkt1,Punkt2]); " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(geom3d):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "point(p1,1,2,3):\npoint(p2 ,4,4,4):\ndsegment(ds1,[p1,p2]):" }}}{PARA 0 "" 0 "" {TEXT -1 63 "Den \+ ersten Punkt verbinden wir nun mit dem gerichteten Segment:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "line(Gerade,[ds1,p1]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 14 "Einfacher Plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "dr1:=d isplay(draw(Gerade),color=green):\ndr2:=display(draw(\{ds1,p1,p2\},sym bol=circle),color=red):\ndisplay(dr1,dr2,orientation=[-45,90]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 9 "Animation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "dr1:=displa y(draw(Gerade),color=green):\ndr2:=display(draw(\{ds1,p1,p2\},symbol=c ircle),color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "displ ay(seq(display(dr1,dr2,axes=none,orientation=[i,i]),i=-180..180),inseq uence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 256 33 "Nachgeschaut - So macht das Maple" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 76 "interface(verboseproc=3):\nprint(geom3d[dsegme nt]);\ninterface(verboseproc=0):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6 %%%dsegG%#P1G%#P2G6)%$segG%#x1G%#x2G%#y1G%#y2G%#z1G%#z2G6#%inCopyright ~(c)~1995~by~Waterloo~Maple~Inc.~All~rights~reserved.G6\"C(@$529#\"\"# 2\"\"$F7-%&ERRORG6#%:wrong~number~of~argumentsG@%/F7F8@$54-%%typeG6$&9 \"6#F8.%%listG0-%$mapG6$-%(readlibG6#%,geom3d/formGFG7$.%(point3dGFU-F <6#%8wrong~type~of~argumentsG@$50-FP6#FGFU0-FP6#&FH6#F:FUFW@'/F7F:C+>8 %--FQ6#%.geom3d/xcoordG6#9%>8&-Fdo6#9&>8'--FQ6#%.geom3d/ycoordGFgo>8(- FapF\\p>8)--FQ6#%.geom3d/zcoordGFgo>8*-FjpF\\p>&8$FR.%+dsegment3dG>&Fb q6#%-geom3d/givenG7$FhoF]p>&Fbq6#%0geom3d/vectformG7%,&Fjo\"\"\"Fbo!\" \",&FepF`rF_pFar,&F^qF`rFhpFarF@-%'RETURNG6#-9!6$9$-%#opGFgoFW--FQ6#%0 geom3d/isassignG6#Fjr-%-setattributeG6$Fjr-%%evalG6#FbqFjrF26$FSFhqF2 " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 262 "" 0 "" {TEXT 256 39 "Gerade durch einen Punkt und eine Ebene" }} {PARA 0 "" 0 "" {TEXT -1 146 "Um eine Gerade durch einen Punkt und ein e Ebene definieren zu k\366nnen, m\374ssen wir uns nat\374rlich erst e inmal einen Punkt und eine Ebene definieren: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(geom3d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "point(p1,4,4,4):\nplane(e1,1*x-2*y+3*z=4,[x,y,z]): " }}}{PARA 0 "" 0 "" {TEXT -1 42 "Den Punkt verbinden wir nun mit der \+ Ebene:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "line(Gerade,[p1,e1 ]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 14 "Einfacher Plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "dr1:=display(draw(Gerade),color=red):\ndr2:=display(draw(\{p1\} ,symbol=circle),color=blue):\ndr3:=display(draw(\{e1\}),color=yellow,s tyle=patchnogrid):\ndisplay(dr1,dr2,dr3,orientation=[-45,90]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 9 "Animation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "dr1:=displ ay(draw(Gerade),color=red):\ndr2:=display(draw(\{p1\},symbol=circle),c olor=blue):\ndr3:=display(draw(\{e1\}),color=yellow,style=patchnogrid) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "display(seq(display(dr 1,dr2,dr3,axes=none,orientation=[i,i]),i=-180..180),insequence=true); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 256 33 "Nach geschaut - So macht das Maple" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "interface(verboseproc=3):\nprint(geom3d[plane]);\ninterface(verb oseproc=0):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6%%$objG%#P1G'%\"nG7%% %nameGF*F*6H%\"xG%\"yG%\"zG%\"pG%$chkG%$disG%$msgG%\"aG%\"bG%\"cG%#coG %#x0G%#y0G%#z0G%$da1G%$da2G%\"vG%#v1G%\"iG%\"jG%(iszerojG%#eqG%#x1G%#y 1G%#z1G%#x2G%#y2G%#o1G%#o2G%#o3G%#z2G%#x3G%#y3G%#z3G%\"sG%#f1G%#f2G%#f 3G6#%inCopyright~(c)~1995~by~Waterloo~Maple~Inc.~All~rights~reserved.G 6\"C)@$529#\"\"#2\"\"$FY-%&ERRORG6#%:wrong~number~of~argumentsG@%/FYFf nC*>8$-%#opG6$\"\"\"9&>8%-Fao6$FZFdo>8&-Fao6$FfnFdo@$2-%%nopsG6#<%F_oF foFjoFfn-Fhn6#%Ccoordinate~names~must~be~differentG>&8'6#%-geom3d/xnam eGF_o>&Fhp6#%-geom3d/ynameGFfo>&Fhp6#%-geom3d/znameGFjo>8(%%trueG@%--% (readlibG6#%6geom3d/isassignednameG6%.F_o.Ffo.FjoC'@$F^pFcp>FgpF_o>F\\ qFfo>F`qFjo>FdqFeq>Fdq%&falseG@+3-%%typeG6$9%.%%listG/-F`p6#F]sFZC%>8G 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FetFj^l/F]hlFh^lC*-Fjal6$F_bl,&FcwFcoFadlFco-Fjal6$Fgcl,&FcwFco-FeblF_ hlFco-F[z6%F`al7%F[tF`blFhclFibl>F_oFafl>FfoFcfl>FjoFefl@$330F_oF`v0Ff oF`v0FjoF`vC%>F]glF_o>FaglFfo>FcglFjoFjfl3Fjr/Fas\"#5C,>87Fco?&86F]sFe q@%1Fd`m\"\"'C$>88-Fj_l6#Ff`m@'/F\\amFeqC$>Fd`m,&Fd`mFcoFcoFco%%nextG/ F\\amF`vC$>Fhu-Fju6$%Bunable~to~determine~if~%a~is~zeroGFf`mF]v-Fhn6#% 8wrong~type~of~argumentsG@%4-F[s6$Ff`m.%*algebraicGF[bm>Fd`mFcam>Fhv-F ao6$\"\"(F]s>F[w-Fao6$\"\")F]s>F^w-Fao6$\"\"*F]s>Fjo-%$mapG6$Fj_l<%F[w FhvF^w@&/Fjo<#FeqF[bm4-%$hasG6$FjoFgr-Fhn6#%;unable~to~define~the~plan eG>FaxFbx>FexF]s>F`y&F]s6#;FhbmF`cm5-F[s6$F]s%(polynomG-F[s6$F]s.%)equ ationGC%@%Fjdm>89,&-Fao6$FcoF]sFco-Fao6$FZF]sF\\y>FaemF]s@$F_]mC'--Fiq 6#%/geom3d/getnameG6&FbxF_oFfoFjo>FgpF_o>F\\qFfo>F`qFjo>FdqFeq@%-F[s6$ Faem-Fidm6$%)anythingGFbpC$>F]_l--Fiq6#%1geom3d/getcoeffsG6$Faem7%F_oF foFjo-Fhy6#-F[z6%F]zF]_lF`gmF[bmF[bm@$Fdq@$-F[dm6$FexFbp-Fhn6#%Othe~co ordinate~name~is~the~same~as~a~parameterG--Fiq6#%0geom3d/isassignG6#F] z-%-setattributeG6$F]z-%%evalG6#FhpF]zFTFTFT" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 315 24 "Gerad e durch zwei Ebenen" }}{PARA 0 "" 0 "" {TEXT -1 115 "Um eine Gerade du rch zwei Ebenen definieren zu k\366nnen, m\374ssen wir uns nat\374rlic h erst einmal zwei Ebenen definieren:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(geom3d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plane(e1,1*x-2*y+3*z=4,[x,y,z]):\nplane(e2,3*x-1*y+5* z=9,[x,y,z]):" }}}{PARA 0 "" 0 "" {TEXT -1 55 "Die beiden Ebenen verbi nden wir jetzt zu einer Geraden:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "line(Gerade,[e1,e2]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 14 "Einfacher Plot" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 204 "dr1:=display(draw(Gerade,thickness =5),color=red):\ndr2:=display(draw(\{e1\}),color=grey,style=patchnogri d):\ndr3:=display(draw(\{e2\}),color=yellow,style=patchnogrid):\ndispl ay(dr1,dr2,dr3,orientation=[-45,90]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 9 "Animation " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "dr1:=display(draw(Gerade,thickness =5),color=red):\ndr2:=display(draw(\{e1\}),color=grey,style=patchnogri d):\ndr3:=display(draw(\{e2\}),color=yellow,style=patchnogrid):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "display(seq(display(dr1,dr2, dr3,axes=none,orientation=[i,i]),i=-180..180),insequence=true);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT 316 23 "Gerade in Parameterform" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(geom3d):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "line(Gerade,[Wert11+Wert12*t , Wert21+Wert22*t, Wert31+Wert31*t ],t ):" }}}{PARA 0 "" 0 "" {TEXT -1 69 "Nun m\374ssen wir uns nat\374rlich noch ein paar konkrete Werte definieren:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Wert11:=1:\n Wert12:=2:\nWert21:=4:\nWert22:=-3:\nWert31:=3:\nWert32:=5:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 256 "" 0 " " {TEXT -1 14 "Einfacher Plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "dr1:= display(draw(Gerade,color=red)):\ndisplay(dr1,orientation=[-45,90]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 " " 0 "" {TEXT -1 9 "Animation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "dr1:=di splay(draw(Gerade),color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "display(seq(display(dr1,axes=none,orientation=[i,i]),i=-180..1 80),insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 317 102 "Gib einen Punkt an, der auf einer gegebenen Gerade n liegt bzw. nicht auf einer gegebenen Geraden liegt" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(geom3d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "point(p1,1,2,3):\npoint(p2,4,4,4):\nline(Ge rade,[p1,p2]):" }}}{PARA 0 "" 0 "" {TEXT -1 69 "Um die Punkte bestimme n zu k\366nnen, brauchen wir die Geradengleichung:" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 23 "gl:=Equation(Gerade,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#glG7%,&\"\"\"F'%\"tG\"\"$,&\"\"#F'F(F+,&F)F'F(F'" }} }{PARA 0 "" 0 "" {TEXT -1 103 "Nun k\366nnen wir f\374r t verschiedene Werte einsetzen und erhalten dann Punkte, die auf der Geraden liegen: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "seq([`F\374r t=`.t.`: `. gl],t=-3..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)7#(%+F|gzr~t=-3:~G7%! \")!\"%\"\"!7#(%+F|gzr~t=-2:~G7%!\"&!\"#\"\"\"7#(%+F|gzr~t=-1:~G7%F/F) \"\"#7#(%*F|gzr~t=0:~G7%F0F5\"\"$7#(%*F|gzr~t=1:~G7%\"\"%F?F?7#(%*F|gz r~t=2:~G7%\"\"(\"\"'\"\"&7#(%*F|gzr~t=3:~G7%\"#5\"\")FE" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 14 "Einfacher Plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "seq(po int(a.t,gl),t=-2..2):\ndr1:=display(draw(\{seq(a.i,i=-2..2)\},symbol=d iamond),color=red):\ndr2:=display(draw(Gerade),color=red):\ndisplay(dr 1,dr2,orientation=[-45,90]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 9 "Animation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 130 "seq(point(a.t,gl),t=-2..2):\ndr1:=display(dra w(\{seq(a.i,i=-2..2)\},symbol=diamond),color=red):\ndr2:=display(draw( Gerade),color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "disp lay(seq(display(dr1,dr2,axes=none,orientation=[i,i]),i=-180..180),inse quence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 53 "Nun ein paar Punkte die nicht auf der Geraden liegen:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "seq([`F\374r t=`.t.`: `,subs (\{t=i,op(1,gl)=i\},gl)],i=-3..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6) 7$%*F|gzr~t=t:~G7%!\"$!\"%\"\"!7$F$7%!\"#F+\"\"\"7$F$7%!\"\"F(\"\"#7$F $7%F(F0\"\"$7$F$7%F,\"\"%F67$F$7%F0\"\"'\"\"&7$F$7%F3\"\")F9" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 256 "" 0 " " {TEXT -1 14 "Einfacher Plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "seq( point(p.i,subs(\{op(1,gl)=i,t=1\},gl)),i=-2..2):\ndr1:=display(draw(\{ seq(a.i,i=-2..2)\},symbol=diamond),color=red):\ndr2:=display(draw(Gera de),color=red):\ndisplay(dr1,dr2,orientation=[-45,90]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 9 "Animation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "wit h(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "seq(point(p.i ,subs(\{op(1,gl)=i,t=1\},gl)),i=-2..2):\ndr1:=display(draw(\{seq(a.i,i =-2..2)\},symbol=diamond),color=red):\ndr2:=display(draw(Gerade),color =red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "display(seq(displ ay(dr1,dr2,axes=none,orientation=[i,i]),i=-180..180),insequence=true); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 256 33 "Nach geschaut - So macht das Maple" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "interface(verboseproc=3):\nprint(geom3d[Equation]);\ninterface(v erboseproc=0):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%$objG6+%\"xG%\"y G%\"zG%#ldG%\"cG%%nochG%\"pG%\"vG%\"tG6#%inCopyright~(c)~1995~by~Water loo~Maple~Inc.~All~rights~reserved.G6\"C&@$529#\"\"\"2\"\"$F7-%&ERRORG 6#%:wrong~number~of~argumentsG@$4-%'memberG6$--%(readlibG6#%,geom3d/fo rmG6#9$<%.%'line3dG.%(plane3dG.%)sphere3dG-F<6#%7wrong~type~of~argumen tG@%/F7F:@%3-%%typeG6$&9\"6#F:.%%nameG/Ffn.%)nochangeG>8)%%trueG>F_o%& falseG>F_oFbo@%/FDFLC&@%/F7F8C$>8,--FF6#%-geom3d/tnameGFI@$/F[p%%FAILG C$--FF6#%/geom3d/getnameG6$FJ.F[p>&-%+attributesGFI6#%/geom3d/indnameG F[pC$@$4F_o>F[q&Fgn6#\"\"#>F[pFdq>8*--FF6#%3geom3d/coordinatesG6#--FF6 #%2geom3d/fixedpointGFI>8+--FF6#%0geom3d/paravectGFI-%$zipG6%R6$F'F(F2 6$%)operatorG%&arrowGF2,&FJF89%F8F2F2F2Fiq7%*&&Fdr6#F8F8F[pF8*&&FdrFeq F8F[p\"\"\"*&&FdrFhnF8F[pFisC'@%35/F7FfqFV/-%%nopsG6#FdqF:C(>8$&FdqFfs >8%&FdqFeq>8&&FdqFhn@$433-FZ6$FgtFin-FZ6$FjtFin-FZ6$F]uFin-F<6#%8wrong ~type~of~argumentsG@$55-%&evalbG6#/FgtFjt-F`v6#/FgtF]u-F`v6#/FjtF]u-F< 6#%Ccoordinate~names~must~be~differentG@$FbqC%>&F\\q6#%-geom3d/xnameGF gt>&F\\q6#%-geom3d/ynameGFjt>&F\\q6#%-geom3d/znameGF]uC&>Fgt--FFF`wFI> Fjt--FFFdwFI>F]u--FFFhwFI@$55/FgtFbp/FjtFbp/F]uFbpC&-Fep6&FJ.Fgt.Fjt.F ]u>F_wFgt>FcwFjt>FgwF]u>8(&F\\q6#%.geom3d/coeffsG@$-%$hasG6$Fdy<%F]uFg tFjt-F<6#%Othe~coordinate~name~is~the~same~as~a~parameterG>8'7,*$)FgtF fqFis*$)FjtFfqFis*$)F]uFfqFis*&FgtF8FjtF8*&FgtFisF]uF8*&FjtFisF]uFisFg tFjtF]uF8/-%(convertG6$-Fjr6%R6$%\"mG%\"nGF2F^sF2*&FJF8FbsF8F2F2F2FeyF az%\"+G\"\"!F26&FgyFawFewFiwF2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 318 43 "Pr\374fe, ob ein Pu nkt auf einer Geraden liegt" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "Es gilt zu untersuchen, ob sich Punkte auf einer Geraden befind en. Hierzu gibt es den Befehl " }{TEXT 256 11 "IsOnObject " }{TEXT -1 2 "im" }{TEXT 257 15 " geom3d-package" }}{PARA 19 "" 0 "" {TEXT -1 73 "IsOnObject(zu_\374berpr\374fender_Punkt, Gerade_auf_der_der_Punkt_lie gen_soll);" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(g eom3d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "point(p1,1,2,3): \npoint(p2,4,4,4):\npoint(p3,-3,4,4):\nline(Gerade,[p1,p2]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "IsOnObject(p1,Gerade);\nIsOn Object(p3,Gerade);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 256 14 "Einfacher Plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "with(plots):\nopt:=symbol=circle:\ndisplay(draw(p1),draw(p2),draw( p3),draw(Gerade),opt);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 256 9 "Animation" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 195 "with(plots):\nopt:=symbol=circle:\ndisplay( seq(display(\ndisplay(draw(p1),opt),\ndisplay(draw(p2),opt),\ndisplay( draw(p3),opt),\ndisplay(draw(Gerade)),\norientation=[i,i]),i=-180..180 ),insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 " " {TEXT 256 33 "Nachgeschaut - So macht das Maple" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "interface(verboseproc=3):\nprint(geom3d[IsOnO bject]);\ninterface(verboseproc=0):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #R6%%\"uG%\"vG'%%condG%%nameG6\"6#%inCopyright~(c)~1995~by~Waterloo~Ma ple~Inc.~All~rights~reserved.GF*C%@$529#\"\"#2\"\"$F1-%&ERRORG6#%:wron g~number~of~argumentsG@$54-%'memberG6$--%(readlibG6#%,geom3d/formG6#9% <%.%'line3dG.%)sphere3dG.%(plane3dG30-F@6#9$.%(point3dG3-%%typeG6$FQ<$ .%%listG.%$setG0-%$mapG6$F@<#-%#opGFP<#FR-F66#%8wrong~type~of~argument sG@%/F?FG-%.geom3d/onobjlG6#9\"-%/geom3d/onobjpsGFfoF*F*F*" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT 319 52 "Bestimme den Abstand eines Punktes von einer Geraden" }}{PARA 0 "" 0 "" {TEXT -1 88 "Um den Abstand eines Punktes zu einer G eraden zu bestimmen, bedient man sich dem Befehl " }{TEXT 256 8 "dista nce" }{TEXT -1 2 ". " }}{PARA 19 "" 0 "" {TEXT -1 24 "distance(Punkt, \+ Gerade);" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(geo m3d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "point(p1,1,2,3):\n point(p2,4,4,4):\npoint(p3,-3,4,4):\nline(Gerade,[p1,p2]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "distance(p3,Gerade);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*&-%%sqrtG6#\"\"&\"\"\"-F&6#\"#9F)#\"\"\"\"\"# " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "`Der Abstand vom Punkt \+ P3 zur Gerade betr\344gt: `,evalf(distance(p3,Gerade),3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%NDer~Abstand~vom~Punkt~P3~zur~Gerade~betr|_ygt :~G$\"$>%!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 256 14 "Einfacher Plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "with(plots):\nopt:=symbol=circle:\nline(Gerade2 ,[p3,p2]):\ndisplay(draw(p1),draw(p2),draw(p3),draw(Gerade),draw(Gerad e2),opt);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 256 9 "Animation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 242 "with(plots):\nopt:=symbol=circle:\nline(Gerade2,[p3, p2]):\ndisplay(seq(display(\ndisplay(draw(p1),opt),\ndisplay(draw(p2), opt),\ndisplay(draw(p3),opt),\ndisplay(draw(Gerade)),\ndisplay(draw(Ge rade2)),\norientation=[i,i]),i=-180..180),insequence=true);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 256 33 "Nachgesch aut - So macht das Maple" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 " interface(verboseproc=3):\nprint(geom3d[distance]);\ninterface(verbose proc=0):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6$%#p1G%#p2G6#%\"fG6#%inC opyright~(c)~1995~by~Waterloo~Maple~Inc.~All~rights~reserved.G6\"C&@$/ 9#\"\"\"@$-%'memberG6$--%(readlibG6#%,geom3d/formG6#9$<$.%*segment3dG. %+dsegment3dG-%'RETURNG6#-9!6#-%#opG6#-&%'geom3dG6#.%*DefinedAsGF:@$0F /\"\"#-%&ERRORG6#%:wrong~number~of~argumentsG>8$-%$mapG6$F6<$F;9%@//FX <#.%(point3dG-%1geom3d/distancepG6$F;Fgn/FX<$F[o.%'line3dG-%3geom3d/di stancepliGF_o/FX<$.%(plane3dGF[o-%2geom3d/distanceplGF_o/FX<#Fbo-%2geo m3d/distanceskGF_o/FX<$FhoFbo-%2geom3d/distancelpGF_o/FX<#Fho-%2geom3d /distanceppGF_o-FT6#%8wrong~type~of~argumentsGF+F+F+" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 5 "Eb ene" }}{PARA 0 "" 0 "Ebene" {TEXT -1 134 "Alles, was man \374ber das A rbeiten mit der Ebene wissen mu\337. Zun\344chst einmal Informationen \+ \374ber die verschiedenen Bildungsm\366glichkeiten:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{SECT 1 {PARA 256 "" 0 "" {TEXT 257 24 "Definition \+ \374ber 3 Punkte" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Mittel s drei verschiedener Punkte l\344sst sich eine Ebene einfach definiere n:" }}{PARA 19 "" 0 "" {TEXT -1 51 "plane(Name_der_Ebene, [Punkt_1, Pu nkt_2, Punkt_3]):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart :with(geom3d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "point(p1, 0,0,0):\npoint(p2,0,2,2):\npoint(p3,4,4,4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plane(ebene,[p1,p2,p3]):" }}}{PARA 0 "" 0 "" {TEXT -1 61 "Die 3 Punkte bilden ein Dreieck, welches sich mit dem Bef ehl " }{TEXT 258 8 "triangle" }{TEXT -1 17 " darstellen l\344\337t." } }{PARA 19 "" 0 "" {TEXT -1 57 "triangle(Name_des_Dreiecks, [Punkt_1, P unkt_2, Punkt_3]):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "triang le(dreieck,[p1,p2,p3]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 20 "Darstellung im Plot:" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 256 14 "Einfacher Plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 226 "with(plots):\nopt:=symbol=circle,color=blue:\ndis play(display(draw(p1),opt),display(draw(p2),opt),display(draw(p3),opt) ,display(draw(ebene),style=patchnogrid,color=coral,display(draw(dreiec k),color=blue)),orientation=[-11,101]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 45 "Beispiel mit zwei Ebenen, die sich schnei den:" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 256 14 "Einfacher Plot" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(geom3d):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 233 "point(p11,0,0,1):\npoint(p1 2,1,1,1):\npoint(p13,1,0,0):\npoint(p21,0,1,1):\npoint(p22,1,1,0):\npo int(p23,1,0,1):\nplane(ebene1,[p11,p12,p13]):\nplane(ebene2,[p21,p22,p 23]):\ntriangle(dreieck1,[p11,p12,p13]):\ntriangle(dreieck2,[p21,p22,p 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{MPLTEXT 1 0 62 "opt1:=symbol=circle,color=blue:\nopt2:=symb ol=circle,color=red:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 395 "di splay(seq(display(\ndisplay(draw(p11),opt1),\ndisplay(draw(p12),opt1), \ndisplay(draw(p13),opt1),\ndisplay(draw(p21),opt2),\ndisplay(draw(p22 ),opt2),\ndisplay(draw(p23),opt2),\ndisplay(draw(ebene1),style=patchno grid,color=grey),\ndisplay(draw(ebene2),style=patchnogrid,color=coral) ,\ndisplay(draw(dreieck1),color=black),\ndisplay(draw(dreieck1),color= grey),\norientation=[i,i]),i=20..70),insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 256 33 "Nachgeschau t - So macht das Maple" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "in terface(verboseproc=3):\nprint(geom3d[plane]);\ninterface(verboseproc= 0):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6%%$objG%#P1G'%\"nG7%%%nameGF* F*6H%\"xG%\"yG%\"zG%\"pG%$chkG%$disG%$msgG%\"aG%\"bG%\"cG%#coG%#x0G%#y 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