Week 
Date of Monday 
Topics 
Sections in ReidSzendrői 
Example sheet 
Comments 
1 
Sep. 29 
Introduction. Metric spaces and examples. Isometric
embeddings and isometries. The CauchySchwarz inequality. 
Appendix A, 1.1 
One 
See the lovely first chapter of Steele's book
for several proofs of the CauchySchwarz inequality.

2 
Oct. 6 
Platonic solids and \(S^2\). The triangle inequality in
\(\EE^n\). Distance versus arclength. Lines, collinearity,
parallelity, and Playfair's axiom. Axiomatic systems and models
of such. 
1.2  1.4 
Two 
All five of the Platonic
solids can be inscribed in \(S^2\), after rescaling.

3 
Oct. 13 
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 
Three 

4 
Oct. 20 
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
area. 
1.11.2  1.16.3 (2.1  2.6) 
Four 
Please read sections 2.1  2.6, on composition of
isometries. 
5 
Oct. 27 
Spherical distance, great circles, polar duals, and angles.
The spherical cosine law and triangle inequality. Classification
of isometries of the twosphere. 
3.1  3.4 
Five 

6 
Nov. 3 
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 
Six 

7 
Nov. 10 
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 
Seven 

8 
Nov. 17 
\(\Isom(\HH^2) \isom O^+(1,2)\). Rotations, translations,
parabolics. Pencils in the three geometries. Elliptic,
parabolic, hyperbolic pencils. Klein model. Independence of the
parallel postulate. 
3.12  3.13 
Eight 

9 
Nov. 24 
Area of hyperbolic triangles. Ideal triangles. Affine space,
subspaces, transformations. The dimension formula, the affine
group. Klein's Erlangen program. Projective equivalence,
projective space. 
3.14, 4.2  4.6, 6.3, 5.2 
Nine 

10 
Dec. 1 
Projective line, projective plane. Dimension formula, the
projective group. Threetransitivity. Desargues's theorem. 
5.3  5.11 
Ten 
