Preface --
1. Straightedge and compass --
1.1. Euclid's construction axioms --
1.2. Euclid'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'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 --
6.1. 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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