MasterMath Geometry 2014 10 / Jeroen Spandaw

Projective topics 1) Perspective drawing 2) Union Jack construction and independence of M of anything but A, B and V. 3) Topology of P1 (circle) and P2: disc glued to Möbius strip along common boundary circle 4) Parabolas and hyperbolas in perspectives; asymptote = tangent at infinity (horizon) 5) Definition of Pn and P(V) 6) P2 as union of R2 and projective line L at infinity using a screen not passing through O 7) Homogeneous coordinates; points, lines and conics in h.c. 8) Duality, dualizing a statement 9) Theorems of Desargues and Pappos. 10) Convenient h.c. in P1 and P2. 11) Projective theorems are equivalent to their perspective views in R2. 12) Going back and forth between P2 and R2. Formulating perspective views of a projective claim. 13) Cross Ratio: 2 definitions (Spandaw and Stillwell), equivalence of these definitions, invariance under projective transformations, invariance under projections from a point. 14) Bijection P → (P, Q ; R, S) between projective line minus {Q, R, S} to R \ {1,0} = P1 \ {1,0,∞} 15) Behaviour of (P, Q ; R, S) under P ↔ Q, R ↔ S, and {P,Q} ↔ {R, S}. 16) Harmonic quadruples, perspective view of midpoint w.r.t. a vanishing point, and Union Jack 17) Conics in P2 and R2: definition, classification and the notion of ‘interesting conic’ 18) Hierarchy of planar geometries: Euclidean – similarity – affine – projective 19) Euclidean, similarity, affine, and projective transformations. In particular, description of an affine transformation R2 → R2 as x → Ax + t, where A is a 2 × 2 – matrix with det(A) ≠ 0. 20) Projectivity of notions ‘point’, ‘line’, ‘conic’, ‘incidence’, ‘tangent’, ‘polar line’, ‘pole’, ‘cross ratio’, ‘harmonic quadruple’, ‘4th harmonic’, ‘interior’ and ‘exterior’ of a conic 21) Prove that a certain notion is affine by giving a definition that involves only projective notions and the line at infinity. 22) Prove that a certain notion is not affine by showing that it is not invariant under a certain affine transformation. 23) Formulating affine perspective views of projective statements concerning points, lines, conics, incidence, tangents, poles, polar lines, and cross ratio. 24) Theorem about relation between polar line and harmonic quadruples. Hyperbolic topics 1) Upper half plane model of the hyperbolic plane: definition of the notions ‘point’, ‘line’, ‘betweenness’, ‘congruence of line segments’, ‘congruence of angles’. 2) Area and defect of triangles 3) Propositions I.1 – I.28 in Euclid hold in hyperbolic plane. 4) Show that a given statement cannot be proved without the parallel axiom by showing it is false in the hyperbolic plane