By Wilhelm Blaschke (auth.)

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Extra resources for Analytische Geometrie

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Die sogenannten konjugierten Quaternionen 0 0 = qo + elql + e2q2 + eaqa, = qo- elql- e2q2- eaqa (8) gehören in (20, 5) zu entgegengesetzten Drehungen: 00 3 = 00 Blaschke Analytische Geometrie. = qij + q~ + q~ + qi = l. (9) 84 I. Grundbegriffe, Vektoren, Matrizen Es gilt die Rechenregel 00'=0'0. (10) Dann läßt sich (1), wenn man = elxl ~* = e1xf ~ + e2x2 + esxs, + e2 x~ + esx: (ll) nimmt, ersetzen durch die Quaternionenformel (12) Eine Quaternion 0 mit 00=1 (13) heißt genormt. Quaternionen werden hier nur wenig verwendet.

K. CLIFFORD 1873 statt Stab < vorgeschlagen hat. Den Ausdruck <

STEINER (1796-1863) hat 1826 für r 1 r 2 = P die Benennung Potenz des Punktes ~zur Kugel ~eingeführt. Man erhält nach {3) diese Potenz, wenn man in die linke Seite der Kugelgleichung (25, 3) die Zeiger von ~ einführt: P(~. l)- a 2 = + x~ + xä + 2a x + 2 a x + 2 a x + a0 • 1 1 2 2 3 {4) 3 Für die Punkte ~ auf ~ ist P = 0, für die Punkte ~ innerhalb ~ ist P < 0, für die Punkte ~ außerhalb ~ist P > 0. Berührt die Gerade g die Kugel ~ im Punkt ~+re, so ist P(~. ~) =r2 gleich dem Quadrat der Tangentenstrecke.

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