Transcription of Poincar´e’s Disk Model for Hyperbolic Geometry
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Poincar´e’s Disk Model for Hyperbolic Geometry
Chapter 9 Poincar e s Disk Model forHyperbolic Saccheri s WorkRecall that Saccheri introduced a certain family of quadrilaterals. Look again at Section remind yourself of the properties of these quadrilaterals. Saccheri studied the threedifferent possibilities for the summit angles of these of the Acute Angle(HAA) The summit angles are acuteHypothesis of the Right Angle(HRA) The summit angles are right anglesHypothesis of the Obtuse Angle(HOA) The summit angles are obtuseSaccheri intended to show that the first and last could not happen, hence he would havefound a proof for Euclid s Fifth Axiom. He was able to show that the Hypothesis of theObtuse Angle led to a contradiction. This result is now know as the Saccheri-LegendreTheorem (Theorem ). He was unable to arrive at a contradiction when he looked atthe Hypothesis of the Acute Angle. He gave up rather than accept that there was anothergeometry available to study. It has been said that he wrote that the Hypothesis of theAcute Angle must be false because God wants it that way.
This result is now know as the Saccheri-Legendre Theorem (Theorem 7.3). He was unable to arrive at a contradiction when he looked at ... Note that this arc is clearly orthogonal to Γ by its construction. ... The hyperbolic trigonometric functions cosh(x) and sinh(x) are defined by: sinh(x) = ex −e−x 2 cosh(x) = ex +e−x 2 and tanh(x ...
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