| Week |
Date of Monday |
Topics |
Sections in Reid-Szendrői |
Example sheet |
Comments |
| 1 |
Oct. 5 |
Introduction. Metric spaces, examples, nonexamples.
Cauchy-Schwarz 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 Cauchy-Schwarz 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
two-sphere. \(\Phi \from \Orth(3) \to \Isom(S^2)\) is
injective. |
3.1 - 3.4 |
Five |
Question asked by a
student. Note that the six extra-ocular
muscles express the three degrees of freedom of an airplane:
pitch, yaw, and roll. This corresponds to the fact that
\(\SO(3)\) is three-dimensional.
|
| 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 time-like. The
hyperbolic plane, the key lemma, hyperbolic distance.
Coordinate-free 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. Three-transitivity,
cross-ratios, distance in Klein model.
|
5.2 - 5.8 |
Ten
Eleven |
Questions asked by
students.
|
