MODERN GEOMETRY II MA 321 Catalogue Description the Three

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MODERN GEOMETRY II MA 321 Catalogue Description The three geometries, parallel postulates and projections, area functions, perpendicular lines and planes, Saccheri quadrilaterals, inversion and reflection, hyperbolic geometry, absolute geometry, Bolyai’s theorem, defects and area, and consistency of the hyperbolic postulates are studied in this course. 3 credits. Prerequisite: Ma 320 Modern Geometry I or Equivalent. Goals A. To enrich a student's knowledge by learning Non-Euclidean Geometries and their implications and consequences. Procedures A. Lecture/Discussion B. Readings and Assigned Problems (Mainly Proofs) C. Research Project Course Content A. Side Trips - Analytic Geometry 1. Introduction 2. Analytic Geometry 3. Historical Perspectives 4. Coordination of the Plane 5. Distance in the Plane 6. Analytic Equations of Straight Lines and Circles 7. Applications of Analytical Geometry B. Side Trips - Transformational Geometry 1. Transformational Geometry 1. Introduction 2. Mappings and Transformations 3. Isometries 4. Applications of Isometries to Theorem Proving 5. Similarities and Their Applications to Theorem Proving 2. Analytic Transformations 1. Introduction 2. Analytical Equations for Isometries 3. Analytical Equations for Similarities 4. Applications of Isometries and Similarities Using Analytical Transformations. 3. Inversion 1. Introduction 2. Inversion in a Circle C. Non-Euclidean Geometries 1. Non-Euclidean Geometries 1. Introduction 2. A Return to Neutral Geometry - The Angle of Parallelism 3. The Hyperbolic Parallel Postulate 4. Some Hyperbolic Results Concerning Polygons 5. Area in Hyperbolic Geometry 6. Showing Consistency - A Model for Hyperbolic Geometry 7. Classifying Theorems 2. Elliptical Geometry 1. A Geometry with no Parallels 2. Two Models 3. Some Results in Elliptic Geometry 3. Geometry in the Real World D. Projective Geometry 1. Introduction 2. The Real Projective Plane 3. Duality 4. Perspectivity 5. Theorem of Desargues 6. Projective Transformations Evaluation Methods 1. 3 Exams. There may be a take home exam due to nature of material - 50% 2. Final 2 hour Comprehensive Exam covering the term - 25% 3. Research Project - 25% Bibliography Required Text: Wallace, Edward C. & West, Stephen F., Roads to Geometry, Prentice Hall, Englewood Cliffs, N.J., 1992. Adler, Claire Fisher, Modern Geometry, An Integrated First Course, 2nd Ed., McGraw Hill Publishing, New York, 1967. Coxeter, H.S.M., Introduction to Geometry, 2nd Ed., John Wiley & Sons, New York, 1969. Eves, Howard A., Survey of Geometry, Rev. Ed., Allyn & Bacon, Boston, Mass., 1972. Fishback, W.T., Projective and Euclidean Geometry, 2nd Ed., John Wiley & Sons, New York, 1969. Greenberg, Marvin Jay, Euclidean and Non Euclidean Geometries, Development and History, 2nd Ed., W.H. Freeman & Co., New York, 1980. Jacobs, Harold R., Geometry, 2nd Ed., W.H. Freeman & Co., New York, 1987. Moise, Edwin E., Elementary Geometry From an Advanced Standpoint, 3rd Ed., Addison-Wesley, New York, 1990. Posamentier, Alfred S., Excursions in Advanced Euclidean Geometry, Rev. Ed., Janson Publishing (Addison-Wesley), Providence, Rhode Island, 1984. (Chapters 1-3) Rich, Barnett, Theory and Problems of Geometry, Schaum's Outline Series, 2nd Ed., McGraw Hill, New York, 1989. Smart, James R., Modern Geometries, 3rd Ed., Brooks/Cole Publishing, California, 1988. Software The Geometric Supposers. Designed by: Education Development Center, Dr. Judah L. Schwartz, MIT/Harvard, and Dr. Michal Yerushalmy, EDC. Available from Sunburst Communications. .

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A Brief Survey of Elliptic Geometry

A Brief Survey of Elliptic Geometry By Justine May Advisor: Dr. Loribeth Alvin An Undergraduate Proseminar In Partial Fulfillment of the Degree of Bachelor of Science in Mathematical Sciences The University of West Florida April 2012 i APPROVAL PAGE The Proseminar of Justine May is approved: ________________________________ ______________ Lori Alvin, Ph.D., Proseminar Advisor Date ________________________________ ______________ Josaphat Uvah, Ph.D., Committee Chair Date Accepted for the Department: ________________________________ ______________ Jaromy Kuhl, Ph.D., Chair Date ii Abstract There are three fundamental branches of geometry: Euclidean, hyperbolic and elliptic, each characterized by its postulate concerning parallelism. Euclidean and hyperbolic geometries adhere to the all of axioms of neutral geometry and, additionally, each adheres to its own parallel postulate. Elliptic geometry is distinguished by its departure from the axioms that define neutral geometry and its own unique parallel postulate. We survey the distinctive rules that govern elliptic geometry, and some of the related consequences. iii Table of Contents Page Title Page ……………………………………………………………………….……………………… i Approval Page ……………………………………………………………………………………… ii Abstract ………………………………………………………………………………………………. iii Table of Contents ………………………………………………………………………………… iv Chapter 1: Introduction ………………………………………………………………………... 1 I. Problem Statement ……….…………………………………………………………………………. 1 II. Relevance ………………………………………………………………………………………………... 2 III. Literature Review …………………………………………………………………………………….