Read e-book online Analytische Geometrie PDF

By Gerd Fischer (auth.)

ISBN-10: 3322964175

ISBN-13: 9783322964175

ISBN-10: 352827235X

ISBN-13: 9783528272357

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Example text

F(po) f(PI) folgt die Behauptung. Übungsau[gabe. Seien Po =(x~O), ... ,x~O», auf einer Geraden gelegen und sei Po iE {I, ... ,n} und es gilt x. TV(Po,PI>P)= x~O) (:)_1(0) Xi Xi *- P I. Dann ist xfO) *- xfl) ftir mindestens ein 2S 1. 2. 5. Wir wollen an sehr einfachen Beispielen erläutern, wie man mit Hilfe von affmen Koordinaten Aussagen der elementaren ebenen Geometrie schnell einsehen kann. 1. Die Diagonalen eines Parallelogramms schneiden sich in den Mittelpunkten. 21). e ~----~ o 2. Die Seitenhalbierenden eines Dreiecks schneiden sich in einem Punkt und teilen sich im Verhältnis 2: 1.

Pn)' Die Skalare Xb ... , Xn heißen die Koordinaten von P bezüglich (Po, ... 19). 'P R2 .. 4. Ein wichtiger Grundbegriff der affrnen Geometrie ist das Teilverhältnis. Sei X ein affrner Raum über K, Y C X eine Gerade und Po, Pb P E Y mit Po PI' Dann ist * T (Y) = K· PoPI C T (X) Es gibt also ein eindeutig bestimmtes A E K mit PoP =APoPI 24 1. Affine Geometrie und wir definieren das Teilverhältnis von Po, PI> P als TV(Po,pl>p):= A Diese Definition kann man noch etwas anders ansehen. (Po, pd ist eine affine Basis von Y.

Setzt man schließlich p p = := f, so ergibt sich es --s1 cos ep und Differentiation ergibt d2 p dep2 + P = 1 es . + Physikalische überlegungen zeigen, daß man die Differentialgleichung des Zweikörperproblems bei einer zu r proportionalen Anziehungskraft auf die Form (*) bringen kann (siehe etwa Berkeley, Physik Kurs 1, § 9, Vieweg 1973). Damit ist gezeigt, daß unsere Kegelschnitte als Bahnkurven auftreten. 50 zeigt einige solcher Kurven, deren Tangente in p senkrecht zur Geraden pq ist. Welche Bahnkurve entsteht, hängt von der Geschwindigkeit im Punkt p ab.

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