MATH 308 Fall 2010 Lecture Summary


Week Date Contents
1 0906
Labour Day
0908 Outline
Introduction: Ceva and Pascal's theorems, Parellel Postulate and non-Euclidean geometry
0910 §1.1.1 Conic sections
§1.1.3 Focus-directrix definition of non-degenerate conics
2 0913 §1.1.4 Focal distance properties of ellipse and hyperbola
§1.1.5 Every conic section is a conic: proof by Dandelin sphere
0915 * Every conic is a conic section
§1.2.1 tangent lines
§1.2.2 Reflection properties (statement)
0917 §1.2.2 proof of reflection properties
§1.3 Recognizing conics
3 0920 §1.3 An example
§1.4.1 Quadric surfaces
0922 §1.4.1 graphing of nondegenerate quadric surfaces in standard forms
§1.4.2 Recognizing quadric surfaces
0924 §1.4.3 a hyperboloid of one sheet is a ruled surface
4 0927 §2.1.1 Isometries and Euclidean transformations of R^2
0929 §2.1.1 examples; isometries and Euclidean transformations of R^2 are the same
1001
§2.1.2 Euclidean transformations of R^2 is a group, examples
Appendix 1. definition and examples of group
§2.1.2 Euclidean congruence of figures
5 1004
§2.1.2 Equivalence relation and equivalence class
§2.2.1 Affine transformations and affine properties
§2.2.2 parallel projection
1006
MT1
1008
§2.2.2 Parallel projections preserve lines, parallel lines and ratio of length along the same line.
§2.2.3 Parallel projections are affine transformations. Each affine transformation is a composition of two parallel projections, and hence also preserves lines, parallel lines and ratio of length along the same line.
6 1011
Thanksgiving
1013
§2.2.3 application to conjugate diameter of ellipses
§2.3.1 finding images of lines
§2.3.2 the affine transformation mapping (0,0), (1,0), (0,1) to any 3 non-collinear points
1015
§2.3.2 The fundamental theorem of affine geometry
§2.3.3 affine transformations preserve signed ratio of lengths on parallel lines
§2.4.2 Ceva's theorem: statement and first proof
7 1018
          Ceva's theorem: application and second proof
§2.4.3 Menelaus theorem: statement, first proof, and application
1020
§2.4.3 second proof
§2.5.1 three affine-congruent classes of non-degenerate conics, center of ellipse/hyperbola is an affine concept
1022
§2.5.1 tangent lines and asymptotes of conics are affine objects
§3.1.1 perspectivity from artist's viewpoint
8 1025
§3.1.2 imagies of lines under perspectivities, vanishing points
§3.1.3 Desargues' theorem
1027
§3.2.1 and §3.2.2 Projective points and lines
1029
§3.2.2 intersection Point of Lines
§3.2.3 embedding planes
9 1101
§3.3.1 projective transformations
§3.3.2 image of Line under proj. transf.
1103
§3.3.2 example
§3.3.3 is skipped
§3.3.4 proj. transf. from reference Points to arbitrary 4 Points
1105
§3.3.4 Fundamental Theorem of Projective Geometry
§3.4.1 Application: proof of Desargues' Theorem
10 1108
§3.4.2 Pappus and Brianchon's theorems, duality
§3.5     cross ratio
1110
MT2
1112
§3.5     examples and theorems
11 1115
§3.5     cross ratio of the intersection points of a pencil of 4 lines with a cutting line is independent of the cutting line; unique fourth point theorem; Pascal's theorem for a circle (statement)
1117
Proof of Pascal's theorem
Chapter 4 sketch: projective congruence of conics, three tangents theorem and Pascal's theorem for conics
1119
§5.1     overview of Chapter 5, inversions
12 1122
§5.1     image of lines, circles, and angles under inversions
1124
§5.2     transformations of the extended complex plane
1126
§5.2     reciprocal and linear functions are composition of inversions
§5.2.4   Stereographic projection: formulas, it maps circles to generalized circles and preserves angles
13
1129
§5.3     inversive transformations, Moebius transformations
1201
Sketch of Non-Euclidean and spherical geometries (Chapters 6 and 7)
1203
review

1214
Final Exam