File:Cross line theorem for perspctivity.svg

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English: The picture illustrates the "cross-line-theorem' in a projective plane for a situation, where 3 points (A1,A2,A3) on a line a are in perspectivity to 3 points (B1,B2,B3) on another line b from a center Z not on a nor b. If the quadrangles A1A2B2B3 and A2A3B4B5 are fano (see de:Fano-Axiom, then the black "crosses" (A1B2,A2B1) and (A2B3,A3B2) have points of intersection F12, F23 NOT on the line ZF (F=ab). The coloured triangles are in perspective from Z, so if Desargues' theorem holds, their correspondig sides intersect on a common line f, which is then a line through F, F12 and F23. Then the perspectivity (central projectivity) with center Z and axis f maps the red triple on a onto the red triple on b and is a projectivity of the whole plane.
Deutsch: Das Bild zeigt eine Desargues-Konfiguration in einer projektiven Ebene, in der das de:Fano-Axiom gilt: Das Punkttripel (A1,A2,A3) auf a ist in perspektiver Lage zu dem Tripel (B1,B2,B3) auf b vom Zentrum Z aus (blaue Zuordnungsgeraden). Die schwarzen "Kreuzlinien" (hier als Strecken gezeichnet) schneiden einander in zwei verschiedenen Punkten, die, wenn das Fano-Axiom gilt, nicht auf der Verbindungsgeraden ZF (F=ab) liegen. Nun liegen auch die bunt gefärbten Dreiecke perspektiv zueinander von Z aus. Also schneiden sich, wenn der de:Satz von Desargues gilt, entsprechende Dreiecksseiten auf einer Achse. einer dieser Schnittpunkte ist F, die anderen beide sind die Kreuzschnittpunkte. Damit hat man die Achse f der ebenen Perspektivität konstruiert, die die perspektive Zuordnung der roten Tripel fortsetzt.
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Author KlioKlein

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current13:17, 6 August 2013Thumbnail for version as of 13:17, 6 August 2013380 × 320 (6 KB)KlioKlein (talk | contribs)User created page with UploadWizard

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