Roads to geometry
著者
書誌事項
Roads to geometry
Prentice Hall, c1998
2nd ed
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注記
Includes bibliographical references (p. 437-439) and index
内容説明・目次
内容説明
Appropriate for junior level college geometry courses. Assumes only a prior course in high school geometry and the mathematical maturity usually provided by a semester of calculus or discrete mathematics.
This book provides a geometrical experience that unifies a mostly Euclidean approach with various non-Euclidean views of the world. It offers the reader a "map" for a voyage through plane geometry and its various branches, as well as side-trips that discuss analytic and transformational geometry.
目次
1. Rules of the Road (Axiomatic Systems).
Historical Background: Axiomatic Systems and their Properties. Finite Geometries. Axioms for Incidence Geometry.
2. Many Ways to Go.
Introduction. Euclid's Geometry and Euclid's Elements. An Introduction to Modern Euclidean Geometries. Hilbert's Model for Euclidean Geometry. Birkhoff's Model for Euclidean Geometry. SMSG Postulates for Euclidean Geometry. Non-Euclidean Geometries.
3. Traveling Together (Neutral Geometry).
Introduction. Preliminary Notions. Congruence Conditions. The Place of Parallels. The Saccheri-Legendre Theorem. The Search for a Rectangle. Summary.
4. One Way to Go (Euclidean Geometry of the Plane).
Introduction. The Parallel Postulate and Some Implications. Congruence and Area. Similarity. Euclidean Results Concerning Circles. Some Euclidean Results Concerning Triangles. More Euclidean Results Concerning Triangles. The Nine-Point Circle. Euclidean Constructions. Summary.
5. Side Trips (Analytic and Transformational Geometry).
Introduction. Analytic Geometry. Transformational Geometry. Analytic Transformations. Inversion. Summary.
6. Other Ways to Go (Non-Euclidean Geometries).
Introduction. A Return to Neutral Geometry: The Angle of Parallelism. The Hyperbolic Parallel Postulate. Hyperbolic Results Concerning Polygons. Area in Hyperbolic Geometry. Showing Consistency: A Model for Hyperbolic Geometry. Classifying Theorems. Elliptic Geometry: A Geometry With No Parallels? Geometry in the Real World. Summary.
7. All Roads Lead To . . . Projective Geometry.
Introduction. The Real Projective Plane. Duality. Perspectivity. The Theorem of Desargues. Projective Transformations. Summary.
Appendix A.
Euclid's Definitions and Postulates Book I.
Appendix B.
Hilbert's Axioms for Euclidean Plane Geometry.
Appendix C.
Birkhoff's Postulates for Euclidean Plane Geometry.
Appendix D.
The SMSG Postulates for Euclidean Geometry.
Appendix E.
Geometer's SketchPad... Scripts for Poincare Model of Hyperbolic Geometry.
Bibliography.
Index.
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