COURSE OUTLINE

I. INTRODUCTION:
 

1.  History, axioms, absolute geometry, 5th postulate, various geometries.

8/26

II. PLANE EUCLIDEAN GEOMETRY
 

1.  An algebraic model of the Euclidean plane: the Gauss plane, Cartesian coordinates.	0.1-0.3	8/28
2.  Complex numbers: the Argand diagram; the conjugate of a complex number; the triangle inequality; trigonometric form of a complex number - De Moivre's Theorem.	0.4-0.8	8/31
3.  Vectors in the Euclidean (and affine) plane; equation of a line; parametric representation of a line.	1.1-1.4  2.1, 3.1	8/28
4.  Algebraic criterion of collinearity of three points (xA+yB+zC=0, where x+y+z=0 and one of x, y, z is non-zero).	3.1-3.3	9/2
5.  Theorem of Menelaus; barycentric coordinates; signed area of a triangle; Theorem of Ceva.	4.1-4.3
6.  Inner product; the nine-point circle.	7.1-7.3
7.  Synthetic method of solving geometrical problems; "locus of points" as a problem solving strategy.
8.  Geometric Construction with a straightedge and compass.

III. GEOMETRICAL TRANSFORMATIONS
 

1.  Geometrical transformations: composition, identity, inverse transformation, notion of a group.	0.11,-0.12
2.  Transformations of the Euclidean plane: translations, rotations, reflections, glide reflections, central  dilations.	41.1-41.5
3.  Complex numbers and inversions of the Euclidean plane.	20.1-20.4
4.  How inversions help to solve some construction problems.	31.1
5.  Isomerties of the Euclidean plane form a group. Every isometry is a collineation.	42.1-42.2,  43.1-43.3,  46.4
6.  Direct and indirect isometries. Fixed points and fixed lines of translations, rotations, reflections, and  glide reflections.	48.1-48.5,  49.1-49.3
7.  Isometries are uniquely determined by images of three points; the classification theorem of isometries of the Euclidean plane.	50.1, 51.1,  51.4
8.  Matrix representation of geometrical transformations.	80.3
9.  Discrete symmetry groups: rosette, frieze, and wallpaper groups.	[21]

IV. MOEBIUS GEOMETRY
 

1.  Inversive plane  as extended Gauss plane; stereographic projection; the group of Moebius transformations.	52.1-52.5  53.1-53.2
2. Moebius transformations map line and circles into lines and circles.	52.3
3.  The cross-ratio; its invariance under Moebius transformations; the reality of the cross-ratio for four distinct points lying on a straight line or a circle.	53.3 -53.5
4.  Conformal property of Moebius transformations;  classification of Moebius transformations.	54.3

V. ELLIPTIC OR SPHERICAL GEOMETRY
 

1.  Incidence geometry of the sphere; distance and the triangle inequality; parametric representation of   lines, perpendicular lines.	{4}
2.  Centroid of a spherical triangle; the area of a spherical triangle.	[11]
3.  Orthogonal transformations of the 3-space; Euler's Theorem; isometries of the sphere, their fixed       points and fixed lines.; spherical trigonometry.	{4}
4. Finite rotation groups.	{4}

VI. PROJECTIVE GEOMETRY
 

1.  Central projection in the 3-dimensional space; projective plane as a set of lines through the origin in the   3-dimensional space.	[13]
2.  Projective plane as an extension of the Euclidean plane by adding "points at "infinity".	[13]
3.   Cross-ratio; the Desargues Theorem and construction with a straightedge in a restricted   domain.	[14]
4.  Principal of duality in projective geometry; dual of the Desargues Theorem.	[13]

VII. HYPERBOLIC OR LOBACHEVSKY GEOMETRY
 

1.  The Poincare and the upper half-plane model of hyperbolic plane.; the parallel axiom; distance in the hyperbolic plane, geodesic lines.	56.1-56.2,  57.1-57.2  58.3
2.  Classification of isometries of the hyperbolic plane; matrix representation of isometries; circles horocycles and equidistant curves.	59.2
3.  Triangles and hyperbolic trigonometry.	59.1
4.  Area of a hyperbolic triangle; the angular defect.	58.1

Plain references  are to Dan Pedoe's "Geometry, a comprehensive course".
References in [  ]  are to Michael Henle's "Modern geometries, the analytic approach"
References in { } are to Patric J. Ryan's "Euclidean and non-Euclidean geometry, an analytic approach"