This video outlines the basic framework of universal hyperbolic geometry---as the projective study of a circle, or later on the projective study of relativistic geometry. Perpendicularity is defined in terms of duality, the pole-polar correspondence introduced by Apollonius, and we explain that the three altitudes of a triangle meet in a point- the orthocenter H. The basic measurements of quadrance and spread in this geometry arise from the cross ratio of suitable points and lines. We state the main formulas: Pythagoras' theorem, the Triple quad formula, Pythagoras' dual theorem, the Triple spread formula, the Spread law and the Cross law and its dual. These are closely related to, but different from the corresponding laws in Rational Trigonometry.CONTENT SUMMARY: notion of perpendicularity @04:48 Perpendicularity via duality @05:33 Do the altitudes of a triangle meet in a point? @10:54 Quadrance: m'ment beween points @15:14 exercise @18:41 remark on Beltrami-Klein model @19:11 Pythagoras' theorem and Triple quad formula @20:30 Spread: m'ment between lines and? quadrance spread duality theorem @23:29 Remark on Beltrami-Klein model @26:45 Pythagoras' dual theorem @28:43 Main formulas for triangles that involve both quadrances and preads @31:13 (THANKS to EmptySpaceEnterprise)
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