||Date of Monday
||Sections in Reid-Szendrői
||Introduction. Metric spaces and examples. Isometric
embeddings and isometries. The Cauchy-Schwarz inequality.
||Appendix A, 1.1
|| See the lovely first chapter of Steele's book
for several proofs of the Cauchy-Schwarz inequality.
||Platonic solids and \(S^2\). The triangle inequality in
\(\EE^n\). Distance versus arc-length. Lines, collinearity,
parallelity, and Playfair's axiom. Axiomatic systems and models
||1.2 - 1.4
||All five of the Platonic
solids can be inscribed in \(S^2\), after rescaling.
||Barycentric coordinates. Isometries preserve barycentric
coordinates, are affine maps. Angles. Polarization identity.
Isometries preserve angles. Orthonormal bases, orthogonal
matrices. Orientation preserving/reversing. Rotations and
reflections in \(\EE^2\).
||1.5 - 1.11.1, Appendix B.1
||Orthogonal complements and the classification of orthogonal
matrices in all dimensions. The parallel postulate and the angle
sum of a euclidean triangle. Classification of isometries of
\(\EE^2\). The cosine law. The sine law and its connection to
||1.11.2 - 1.16.3 (2.1 - 2.6)
||Please read sections 2.1 - 2.6, on composition of
||Spherical distance, great circles, polar duals, and angles.
The spherical cosine law and triangle inequality. Classification
of isometries of the two-sphere.
||3.1 - 3.4
||Spherical area, lunes, and Girard's lemma. Hyperbolic trig,
Lorentz dot product, the hyperbolic plane. Hyperbolic distance,
angles, unit tangent vectors.
||3.5 - 3.9
||Lorentzian orthogonals, lines parametrized by arclength,
characterizing unit tangents. Cosine law and the triangle
inequality. Semidirect products, \(\Isom(\EE^2) \isom \RR^2 \semi
O(2)\), \(\Isom(S^2) \isom O(3)\).
||3.10 - 3.11
||\(\Isom(\HH^2) \isom O^+(1,2)\). Rotations, translations,
parabolics. Pencils in the three geometries. Elliptic,
parabolic, hyperbolic pencils. Klein model. Independence of the
||3.12 - 3.13
||Area of hyperbolic triangles. Ideal triangles. Affine space,
subspaces, transformations. The dimension formula, the affine
group. Klein's Erlangen program. Projective equivalence,
||3.14, 4.2 - 4.6, 6.3, 5.2
||Projective line, projective plane. Dimension formula, the
projective group. Three-transitivity. Desargues's theorem.
||5.3 - 5.11