Fundamental Concepts of Geometry (eBook)

Bruce E. Meserve (Autor)

eBook Download: EPUB

2014
336 Seiten
Dover Publications (Verlag)
978-0-486-15226-4 (ISBN)

Infos

Demonstrates relationships between different types of geometry. Provides excellent overview of the foundations and historical evolution of geometrical concepts. Exercises (no solutions). Includes 98 illustrations.

---

Fundamental Concepts of Geometry demonstrates in a clear and lucid manner the relationships of several types of geometry to one another. This highly regarded work is a superior teaching text, especially valuable in teacher preparation, as well as providing an excellent overview of the foundations and historical evolution of geometrical concepts. Professor Meserve (University of Vermont) offers students and prospective teachers the broad mathematical perspective gained from an elementary treatment of the fundamental concepts of mathematics. The clearly presented text is written on an undergraduate (or advanced secondary-school) level and includes numerous exercises and a brief bibliography. An indispensable taddition to any math library, this helpful guide will enable the reader to discover the relationships among Euclidean plane geometry and other geometries; obtain a practical understanding of "e;proof"e;; view geometry as a logical system based on postulates and undefined elements; and appreciate the historical evolution of geometric concepts.

Inhalt

CHAPTER 1. FOUNDATIONS OF GEOMETRY1-1 Logical systems1-2 Logical notations1-3 Inductive and deductive reasoning1-4 Postulates1-5 Independent postulates1-6 Categorical sets of postulates1-7 A geometry of number triples1-8 Geometric invariantsCHAPTER 2. SYNTHETIC PROJECTIVE GEOMETRY2-1 Postulates of incidence and existence2-2 Properties of a projective plane2-3 Figures2-4 Duality2-5 Perspective figures2-6 Projective transformations2-7 Postulate of Projectivity2-8 Quadrangles2-9 Complete and simple n-points2-10 Theorem of Desargues2-11 Theorem of Pappus2-12 Conics2-13 Theorem of Pascal2-14 SurveyCHAPTER 3. COORDINATE SYSTEMS3-1 Quadrangular sets3-2 Properties of quadrangular sets3-3 Harmonic sets3-4 Postulates of Separation3-5 Nets of rationality3-6 Real projective geometry3-7 Nonhomogeneous coordinates3-8 Homogeneous coordinates3-9 SurveyCHAPTER 4. ANALYTIC PROJECTIVE GEOMETRY4-1 Representations in space4-2 Representations on a plane4-3 Representations on a line4-4 Matrices4-5 Cross ratio4-6 Analytic and synthetic geometries4-7 Groups4-8 Classification of projective transformations4-9 Polarities and conics4-10 Conics4-11 Involutions on a line4-12 SurveyCHAPTER 5. AFFINE GEOMETRY5-1 Ideal points5-2 Parallels5-3 Mid-point5-4 Classification of conics5-5 Affine transformations5-6 Homothetic transformations5-7 Translations5-8 Dilations5-9 Line reflections5-10 Equiaffine and equiareal transformations5-11 SurveyCHAPTER 6. EUCLIDEAN PLANE GEOMETRY6-1 Perpendicluar lines6-2 Similarity transformations6-3 Orthogonal line reflections6-4 Euclidean transformations6-5 Distances6-6 Directed angles6-7 Angles6-8 Common figures6-9 SurveyCHAPTER 7. THE EVOLUTION OF GEOMETRY7-1 Early measurements7-2 Early Greek influence7-3 Euclid7-4 Early euclidean geometry7-5 The awakening in Europe7-6 Constructions7-7 Descriptive geometry7-8 Seventeenth century7-9 Eighteenth century7-10 Euclid's fifth postulate7-11 Nineteenth and twentieth centuries7-12 SurveyCHAPTER 8. NONEUCLIDEAN GEOMETRY8-1 The absolute polarity8-2 Points and lines8-3 Hyperbolic geometry8-4 Elliptic and spherical geometries8-5 ComparisonsCHAPTER 9. TOPOLOGY9-1 Topology9-2 Homeomorphic figures9-3 Jordan Curve Theorem9-4 Surfaces9-5 Euler's Formula9-6 Tranversable networks9-7 Four-color problem9-8 Fixed-point theorems9-9 Moebius strip9-10 SurveyBIBLIOGRAPHYINDEX

Produkt-Details

• Erscheint lt. Verlag: 8.12.2014
• Reihe/Serie: Dover Books on Mathematics
• Sprache: englisch
• Maße: 140 x 140 mm
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Schlagworte: advanced secondary school • Bibliography • Complex • Concepts • euclidean plane geometry • Exercises • fundamentals • Geometry • graduate • Guide • historical evolution • Math • mathematical perspective • Mathematics • Overview • Postulates • practical understanding • school • Teacher Preparation • teaching text • Textbook • undefined elements • undergraduate • university of vermont
• ISBN-10: 0-486-15226-X / 048615226X
• ISBN-13: 978-0-486-15226-4 / 9780486152264