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Non-Euclidean Geometry Skyler W The University of Maine DigitalCommons@UMaine Electronic Theses and Dissertations Fogler Library 2000 Non-Euclidean Geometry Skyler W. Ross Follow this and additional works at: http://digitalcommons.library.umaine.edu/etd Part of the Geometry and Topology Commons Recommended Citation Ross, Skyler W., "Non-Euclidean Geometry" (2000). Electronic Theses and Dissertations. 426. http://digitalcommons.library.umaine.edu/etd/426 This Open-Access Thesis is brought to you for free and open access by DigitalCommons@UMaine. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of DigitalCommons@UMaine. NON-EUCLIDEAN GEOMETRY By Skyler W. Ross B.S. University of Maine, 1990 A THESIS Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Arts (in Mathematics) The Graduate School University of Maine May, 2000 Advisory Committee: William O. Bray: Chair and Professor of Mathematics, Co-Advisor Eisso J. Atzema: Instructor of Mathematics, Co-Advisor Robert D. Franzosa: Professor of Mathematics Henrik Bresinsky: Professor of Mathematics Acknowledgments The Author would like to express his gratitude to the members of the thesis advisory committee for their time, effort and contributions, to Dr. Grattan Murphy who first introduced the author to non-Euclidean geometries, and to Jean-Marie Laborde for his permission to include the demonstration version of his software, Cabri II, with this thesis. Thanks also to Euclid, Henri Poincaré, Felix Klein, Janos Bolyai, and all other pioneers in the field of geometry. And thanks to those who wrote the texts studied by the author in preparation for this thesis. A special debt of gratitude is due Dr. Eisso J. Atzema. His expert guidance and critique have made this thesis of much greater quality than the author could ever have produced on his own. His contribution is sincerely appreciated. ii Table of Contents Acknowledgments……….………………………………………………..………………ii List of Figures ...................................................……...........…..........................................vi List of Theorems and Corollaries....…...............................................................................xi Chapter I: The History of Non-Euclidean Geometry ….....................................................1 The Birth of Geometry .....................................……...............................................1 The Euclidean Postulates........................................….............................................2 The Search for a Proof of Euclid’s Fifth ....................….........................................3 The End of the Search ...................................................……..................................6 A More Complete Axiom System .......................................…................................7 Chapter II: Neutral and Hyperbolic Geometries …............................…..........................11 Neutral Geometry .......................................................................……...................11 Hyperbolic Geometry ..........................................................................……..........23 Saccheri and Lambert quadrilaterals .............…........................................26 Two kinds of hyperbolic parallels ....................….....................................28 The in-circle and circum-circle of a triangle .......…..................................41 Chapter III: The Models ...............................................................……............................44 The Euclidean Model ..............................................................…..........................44 The Klein Disk Model .................................................................….....................45 The metric of KDM .........................................................…….................47 Angle measure in KDM .......................................................…….............47 The Poincaré Disk Model .....................................................................…............50 The metric of PDM .....................................................................…....…..51 Angle measure in PDM..................................................................……....51 iii The Upper Half-Plane Model .............…...............................................................53 Angle measure in UHP ...............…...…......................……….................56 The metric of UHP .........................…......................................….............57 The hyperbolic postulates in UHP .....…......................................….........64 Chapter IV: Isometries on UHP ...................................……............…............................66 Isometries on the Euclidean Plane ..........................…..............……....................66 Reflection .............................................…......……...............…................66 Translation ...............................................….........…..........…..................67 Rotation .......................................................…..........……....…................68 Glide-reflection .............................................…..............…....…..............70 Euclidean inversion ..........................................….....................…............72 Isometries on UHP ...........................................................….....................……....79 Reflection .............................……............................….............................79 Rotation ......................................……..........................….........................81 º-Rotation ........................................……........................…….................82 Translation ..............................................….............................….............83 Glide-reflection ...........................................…............................…..........85 Chapter V: Triangles in UHP .................................................…......................................86 Angle Sum and Area .......................................................…..................................87 Trigonometry of the Singly Asymptotic Right Triangle .....…..............................89 Trigonometry of the General Singly Asymptotic Triangle ...…............................91 Trigonometry of the Right Triangle .........................................….........................93 Trigonometry of the General Triangle ........................................…......................97 Chapter VI: Euclidean Circles in UHP .....................................................….................101 Hypercycles ........................................................................................…….........101 Circles .................................................……........................................................103 Horocycles ................................................……..................................................105 iv Chapter VII: The Hyperbolic Circle ..........................…................................................109 The Hyperbolic and Euclidean Center and Radius…..........................................109 Circumference .........................................................……....................................110 Area ...............................................................................……..............................111 The Limiting Case ...............................................................…............................113 Hyperbolic P .........................................................................…….....................114 The Angle Inscribed in a Semicircle ...........................................…....................114 Chapter VIII: In-Circles and Circum-Circles ............................................…................116 In-Circles ..........................................................................................….......…....116 The in-circle of the ordinary triangle ....................................…..……....116 The in-circle of the asymptotic triangle ....................................……......117 Circum-Circles ................................................................................….......….....122 References ................................................................……......................................….....130 Appendix: Constructions ...............................................…......................................…..131 Constructions in Euclidean Space ..........................….........................................131 Constructions in KDM ...............................................….....................................133 Constructions in PDM ....................................................…….............................139 Constructions in UHP ...........................................................…..........................145 Biography of the Author .......................................................................…......................151 v List of Figures Figure 1.1 Legendre’s ‘proof’ of the parallel postulate...............….................................4 Figure 1.2 Bolyai’s ‘proof’ of the parallel postulate......................…...............................5 Figure 2.1 Congruent alternate interior angles implies parallelism..…...........................12 Figure 2.2 The external angle is greater than either remote interior
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