## MATH 308 Fall 2011 Lecture Summary

 Week Date Contents 1 0905 Labour Day 0907 outline, Euclidean geometry, overview and examples (Ceva and Pascal theorems, Apollonian circles, angular excess on sphere, angular defect on Poincare disk) L1 0909 Chapter 1. Conic sections §1.1.1 Conics §1.1.3 Focus-directrix definition of non-degenerate conics L2 2 0912 §1.1.4 focal distance properties §1.1.5 conic sections are conics, proof by Dandelin sphere L3 0914 §1.2.1 tangent lines §1.2.2 reflection properties of conic mirrors L4 0916 §1.2.2 continue §1.3   examples that quadratic curves are conics L5 3 0919 §1.3   writing a quadratic equation in standard form L67 0921 §1.3   finding the axes and vertices of a quadratic equation L67 0923 Section 2.1 Euclidean geometry §2.1.1 What is Euclidean geometry? isometries, rotations L8 4 0926 §2.1.1 reflections, Euclidean transformations; group L9 0928 §2.1.1 examples of group properties of Euclidean transformations§2.1.2 Euclidean congruence Chapter 2. Affine geometry §2.2.1 affine transformations L10 0930 §2.2.1 affine properties and affine geometry §2.2.2 parallel projections L11 5 1003 §2.2.2 a parallel projection preserves lines, parallel lines, and ratio of length along same direction §2.2.3 a parallel projection is an affine transformation; any affine transformation is a composition of two parallel projections L12 1005 MT1 1007 §2.2.3 application to conjugate diameters of ellipses §2.3.1 finding the equation of the image of a line under an affine transformation L13 6 1010 Thanksgiving 1012 §2.3.2 Fundamental Theorem of affine geometry §2.3.3 algebraic proof of basic affine properties; signed ratio L14 1014 §2.3.3 more on signed ratio §2.4.2 Ceva's theorem L15 7 1017 §2.4.3 Menelaus' theorem and example L16 1019 §2.5.1 conics have exactly 3 equivalent classes §2.5.2 examples of affine properties of conics L17 1021 §2.5.2 more examples Chapter 5. inversive geometry: overview§5.1 inversions L18 8 1024 Formula for inversion, images of lines and circles, a generalized circle is mapped to a generalized circle L19 1026 angles and parallel lines under inversion L20 1028 §5.2 review complex plane; isometries on complex plane: notation, as compositions of reflections L21 9 1031 scaling, inversion, linear and reciprocal functions, extended complex plane, stereographic projection L22 1102 stereographic projection preserves circles and magnitude of angles §5.3 Inversive transformations, properties, and geometry. Moebius transformation. L23 1104 composition and inverse of Moebius transformations, associated matrices, the Moebius group, an inversive transformation is either M(z) or M(\bar z) L24 10 1107 §5.4 Fundamental theorem of inversive geometry L25 1109 MT2 1111 Remembrance Day 11 1114 how to decide whether 4 points belong to the same circle §5.5 Coaxal families of cirlces. Appolonian circles, algebraic proof. L26 1116 geometric proof, Coaxal circles theorem, relation between inverse points and Appolonian circles L27 1118 continue the proof of the lemma. Inverse points are preserved under inversive transformations. Concentricity theorem. Note we skip §5.5.4 L28 12 1121 Chapter 6. Non-Euclidean Geometry. Recall Euclidean Geometry. §6.1 Poincare disk, d-points and d-lines. L29 1123 Non-Euclidean reflections and transformations, Origin Lemma, Axiom 5 L30 1125 more examples§6.2 formula of Non-Euclidean transformations§6.3 distance function L31 13 1128 §6.3 distance function continued, midpoint, circle L32 1130 A non-Euclidean circle is a Euclidean circle in D examples of uncovered results L33 1202 review 1208 Final Exam