Week 
Date of Monday 
Topics 
Sections in ReidSzendrői 
Example sheet 
Comments 
1 
Oct. 5 
Introduction. Metric spaces, examples, nonexamples.
CauchySchwarz inequality. Barycentric coordinates. The triangle
inequality in \(\EE^n\). 
Appendix A, 1.1 
One 
See the lovely first chapter of Steele's book
for several proofs of the CauchySchwarz inequality.
There are no support classes this week.

2 
Oct. 12 
Lines, collinearity, parallelity, barycentric coordinates.
Parallel postulate, Playfair's axiom. Axiomatic systems and
models of such. Isometric embeddings and isometries. Klein's
Erlangen program. Angles. Isometries of \(\EE^n\) preserve
barycentric coordinates. Isometries preserve barycentric
coordinates, are affine maps.

1.2  1.9 
Two 

3 
Oct. 19 
Isometries preserve angles. Polarization identity.
Orthonormal bases, orthogonal matrices. Orientation
preserving/reversing. Rotations and reflections in \(\EE^2\).
Orthonormal frames, coordinate systems. Orthogonal complements
and the classification of orthogonal matrices in all dimensions.

1.10  1.13, Appendix B.1  B.3 
Three 

4 
Oct. 26 
Classification of isometries of \(\EE^2\). The parallel
postulate and the angle sum of a euclidean triangle. Cosine,
sine. Cosine law, sine law, area. Sphere, great circles,
antipodal points. 
1.14  1.16.3, 2.1  2.6, 3.1 
Four 
Questions asked by
students.
Sections 2.1  2.6 will not be covered in lecture  please read
these over the weekend.
We used area to prove the law of sines. However, giving a
rigorous definition of area
is not easy.

5  Nov. 2 
Spherical distance, chordal distance. Spherical angles,
spherical triangles. First spherical cosine law and triangle
inequality. Polar duals and second cosine law. Isometries of the
twosphere. \(\Phi \from \Orth(3) \to \Isom(S^2)\) is
injective. 
3.1  3.4 
Five 
Question asked by a
student. Note that the six extraocular
muscles express the three degrees of freedom of an airplane:
pitch, yaw, and roll. This corresponds to the fact that
\(\SO(3)\) is threedimensional.

6 
Nov. 9 
\(\Phi \from \Orth(3) \to \Isom(S^2)\) is surjective.
Spherical isometries, area, and Girard's lemma. Hyperbolic trig,
Lorentz dot product. Space, light, and timelike. The
hyperbolic plane, the key lemma, hyperbolic distance.
Coordinatefree definition of angle. 
3.5  3.9 
Six 
Questions asked by
students.

7 
Nov. 16 
Lorentzian orthogonals, mixed planes, great hyperbolas.
"Unit tangent vectors", lines parametrized by arclength. Cosine
law and the triangle inequality. Semidirect products,
\(\Isom(\EE^2) \isom \RR^2 \semi \Orth(2)\), \(\Isom(S^2) \isom
\Orth(3)\). 
3.10  3.11 
Seven 
Questions asked by
students.

8 
Nov. 23 
\(\Isom(\HH^2) \isom \Orth^+(1,2)\). Rotations, parabolics,
translations. Pencils in the three geometries. Elliptic,
parabolic, hyperbolic pencils. Klein model. 
3.12  3.13 
Eight 
Questions asked by
students.

9 
Nov. 30 
Independence of the parallel postulate. Area of hyperbolic
triangles. Ideal triangles. Affine space, affine subspaces,
transformations, the affine group. The dimension formula, Klein's
Erlangen program. 
3.14, 4.2  4.6, 6.3 
Nine 

10 
Dec. 7 
Projective equivalence, projective space. Projective lines,
the projective plane. Dimension formula, the projective group.
Fundamental theorem of projective geometry. Threetransitivity,
crossratios, distance in Klein model.

5.2  5.8 
Ten
Eleven 
Questions asked by
students.
