\mnb150ÿ{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\froman\fcharset1 Times New Roman;}{\f4\fmodern\fprq1 Courier New;}{\f5\fmodern\fprq1\fcharset1 Courier New;}} {\colortbl\red0\green0\blue0;\red255\green0\blue0;\red0\green0\blue128;} \deflang1031\pard\ri4\plain\f3\fs26\cf0\b Affine Abbildungen mittels Matrizen\plain\f3\fs26\cf0 \par \plain\f3\fs22\cf0 \par Eine Ausgangsfigur B wird als geschlossener Streckenzug eingegeben. Die Eingabe der Punkte erfolgt als Vektoren \par in Matrixformat. B entsteht dann als "Hintereinanderschaltung" der Vektoren. Diese Originalfigur wird \par mit Hilfe einer Matrix A in eine Bildfigur T verwandelt. Dabei gilt T = A*B (Matrizenmultiplikation). \par Die Darstellung im KOS erfolgt anschlie\'dfend in drei Schritten: \par \tab \par \tab 1) Zun\'e4chst werden die Matrixwerte von Original und Bild in MuPAD-Punkte \'fcbersetzt. \par \tab 2) Diese Punkte werden dann als Polygon interpretiert. \par \tab 3) Die beiden Polygone werden gezeichnet. \par \par Die Abbildungsmatrix befindet sich am Ende des Programms, da auf diese Weise Matrix und bildliche Darstellung \par gemeinsam auf dem Bildschirm sichrbar sind. Das erleichtert das Experimentieren mit der Matrix. \par \par \par \plain\f3\fs22\cf0\b Die Eingabe der Vektoren und deren "Hintereinanderschaltung" zur Originalfigur B: \par \plain\f3\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf1 {\pntext\f1\'b7\tab}P1:= matrix([1.5,1]): \par \pard\li600\ri1\fi-300\plain\f4\fs22\cf1 P2:= matrix([2,1]): \par P3:= matrix([2,3]): \par P4:= matrix([3,3]): \par P5:= matrix([3,4]): \par P6:= matrix([2,4]): \par P7:= matrix([2,5]): \par P8:= matrix([4,5]): \par P9:= matrix([4,6]): \par P10:= matrix([1.5,6]): \par P11:= matrix([1.5,1]): \par B:= P1.P2.P3.P4.P5.P6.P7.P8.P9.P10.P11 \par \pard\li50\ri6\plain\f4\fs22\cf2\protect {\pict\wmetafile8\picw6255\pich1143\picwgoal3546\pichgoal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}\plain\f4\fs22\cf2\protect \par \pard\li50\ri2\plain\f4\fs22\cf2\protect \par \pard\ri4\plain\f3\fs22\cf0 \par \plain\f3\fs22\cf0\b 1) \'dcbersetzung der Vektoren in Punkte (MuPAD plottet sonst nicht): \par \par \plain\f3\fs22\cf0 Zur Logik des Befehls: \plain\f5\fs22\cf0 plot::Point\plain\f3\fs22\cf0 erwartet eine x- und eine y-Koordinate. Diese Werte werden geliefert, \par indem spaltenweise erst der 1. Zeilenwert und dann der 2. Zeilenwert eingelesen werden.\plain\f3\fs22\cf0\b \par \plain\f3\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf1 {\pntext\f1\'b7\tab}Orig_proto:=plot::Point(linalg::col(B, i)[1],linalg::col(B, i)[2]) $ i = 1..11 \par \pard\li50\ri6\plain\f4\fs22\cf2\protect {\pict\wmetafile8\picw-30958\pich827\picwgoal19603\pichgoal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}\plain\f4\fs22\cf2\protect \par \pard\li50\ri2\plain\f4\fs22\cf2\protect \par \pard\ri4\plain\f3\fs22\cf0\b 2) Interpretation der Punkte als Polygonzug: \par \plain\f3\fs22\cf0 \par \plain\f5\fs22\cf0 Closed=TRUE\plain\f3\fs22\cf0 sorgt daf\'fcr, dass das Ploygon durchgezogenen Strecken und nicht nur die Eckpunkte erh\'e4lt. \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf1 {\pntext\f1\'b7\tab}Orig:=plot::Polygon( \par \pard\li600\ri1\fi-300\plain\f4\fs22\cf1 Orig_proto, \par Closed = TRUE \par ) \par \pard\li50\ri6\plain\f4\fs22\cf2\protect {\pict\wmetafile8\picw2983\pich827\picwgoal1691\pichgoal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}\plain\f4\fs22\cf2\protect \par \pard\li50\ri2\plain\f4\fs22\cf2\protect \par \pard\ri4\plain\f3\fs22\cf0 HIER m\'fcsste nun die Matrix A definiert werden. Aus o.g. Gr\'fcnden findet sie sich aber \par direkt unter der Bildausgabe. \par \par \plain\f3\fs22\cf0\b \par Berechnung der Bildfigur:\plain\f3\fs22\cf0 \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf1 {\pntext\f1\'b7\tab}T:= A*B \par \pard\li50\ri6\plain\f4\fs22\cf2\protect {\pict\wmetafile8\picw10340\pich1143\picwgoal5862\pichgoal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