Martin Flashman's Courses - Math 371 Spring, '03

Geometry Notes

[Work in Progress DRAFT VERSION 4-7-03]

1/22 Introduction
1/24 Continue discussion of what is "geometry"? 
Start on The Pythagorean Theorem
1/27 The Pythagorean Theorem
1/29  Dissections
1/31 Finish Dissections. 
Begin M&I's Euclidean Geometry
2/3 Constructions
2/5 More on Constructions and the real number line.
2/7 The real number line.
 2/10 Isometries
2/12 Classification of Isometries & More :)   2/14 More on Isometries.
2/17 Symmetry, Similarity & Proportion
2/19 Similarity and proportion
2/21 More on Proportion and Measurement
2/24Inversion and Orthogonal Circles 
2/26 Inversion and Beginning to See The Infinite. 2/28 More on seeing the infinite.
3/3 The Affine Line and Homogeneous Coordinates.
3/5  More on The Affine Line and Homogeneous Coordinates and the Affine Plane 
3/7 More on The Affine Lineand Homogeneous Coordinates and the Affine Plane 
3/10 Axioms and Finite geometries!
3/12 Connecting Axioms to Models. Introduction to projective geometry with homogeous coordinates.
  3/14  Z2 and  Finite Projective Geometry. 
Video Orthogonal Projection.
3/17 No class Spring Break!
3/19 No class Spring Break!
3/21No class Spring Break!
3/24 Review of Algebraic and Visual Models for Affine and Projective Geometries. 
Introduction to Desargues' Theorem- a result of projective geometry.
3/26 More on Visual and algebraic models for plane geometry. More axioms for synthetic projective plane geometry. 
3/28 Axioms for Synthetic Projective Geometry (see M&I)
3/31 No Class C. C. Day 
4/2 More examples of proofs in synthetic projective geometry. Proof of Desargues' Theorem in the Plane. 
4/4 A look at duality and some applications.
4/7  Conics. Introduction to Pascal's Theorem.
4/9  Sections and Perspectively related Figures
4/11 Some key duality, perspective reconsidered.
4/14  Projective relations, Start Isometries with Homog. Coord.
4/16 Projective Line transformations: Synthetic Projectivities; Matrices
4/18 Quiz #2 More on Matrix Projective Transformations.
4/21 Harmonics: equivalences and construction.
4/23 Harmonics: uniqueness and coordinates for Projective Geometry. 
Planar transformations and Matrices
4/25 A Non-Euclidean Universe and Inversions.

Case 1
Case 2
Case 3. The vertex is vk with k different from 3 or n. Then consider the polygonal regions Q3 = v1v2...vk which has k vertices (k<n) and Q4 = which has n-(k-2)<n vertices. By induction Q3 and Q4 can be triangulated, so the original polygon is triangulated using the triangulations of Q3 and Q4.
Case 3


SPRING BREAK-- Check the assignments page for the work due on Thursday, March 27th.

3/26 Reconsidered how we model geometry visually and algebraically:
Points:Ordinary coordinates(x,y) 
Lines: Ax+By+C=0
Points: Ordinary (x,y) or generally homogeneous <x,y,z> x,y,z not all 0. 
Lines: Ax+By+C=0 or Ax+By+Cz = 0 or [A,B,C]
Points: <x,y,z> x,y,z not all 0. 
Lines:  Ax+By+Cz = 0 or [A,B,C] A,B,C not all 0.