内容简介
作者简介
John Stillwell is Professor of Mathematics at the University of San Francisco. He is the author of several highly regarded books published by Springer, including Elements of Number Theory 2003, Mathematics and Its History Second Edition, 2002, Numbers and Geometry 1998 and Elements of Algebra 1994.
本书目录
Preface --1. Straightedge and compass --
1.1. Euclid&39;s construction axioms --
1.2. Euclid&39;s construction of the equilateral triangle --
1.3. Some basic constructions --
1.4. Multiplication and division--
1.5. Similar triangles --
1.6. Discussion --
2. Euclid&39;s approach to geometry --
2.1. The parallel axiom --
2.2. Congruence axioms --
2.3. Area and equality --
2.4. Area of parallelograms and triangles --
2.5. The Pythagorean theorem --
2.6. Proof of the Thales theorem --
2.7. Angles in a circle --
2.8. The Pythagorean theorem revisited --
2.9. Discussion --
3. Coordinates --
3.1. The number line and the number plane --
3.2. Lines and their equations --
3.3. Distance --
3.4. Intersections of lines and circles --
3.5. Angle and slope --
3.6. Isometries --
3.7. The three reflections theorem --
3.8. Discussion --
4. Vectors and euclidean spaces --
4.1. Vectors --
4.2. Direction and linear independence --
4.3. Midpoints and centroids --
4.4. The inner product --
4.5. Inner product and cosine --
4.6. The triangle inequality --
4.7. Rotations, matrices, and complex numbers --
4.8. Discussion.
5. Perspective --
5.1. Perspective drawing --
5.2. Drawing with straightedge alone --
5.3. Projective plane axioms and their models --
5.4. Homogeneous coordinates --
5.5. Projection --
5.6. Linear fractional functions --
5.7. The cross-ratio --
5.8. What is special about the cross-ratio? --
5.9. Discussion --
6. Projective planes --
6.1. Pappus and Desargues revisited --
6.2. Coincidences --
6.3. Variations on the Desargues theorem --
6.4. Projective arithmetic --
6.5. The field axioms --
6.6. The associative laws --
6.7. The distributive law --
6.8. Discussion --
7. Transformations --
7.1. The group of isometries of the plane --
7.2. Vector transformations --
7.3. Transformations of the projective line --
7.4. Spherical geometry --
7.5. The rotation group of the sphere --
7.6. Representing space rotations by quaternions --
7.7. A finite group of space rotations --
7.8. The group S³ and RP³ --
7.9. Discussion --
8. Non-Euclidean geometry --
8.1. Extending the projective line to a plane --
8.2. Complex conjugation --
8.3. Reflections and Mobius transformations --
8.4. Preserving non-Euclidean lines --
8.5. Preserving angle --
8.6. Non-Euclidean distance --
8.7. Non-Euclidean translations and rotations --
8.8. Three reflections or two involutions --
8.9. Discussion --
References --
Index.
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