Notes on Geometry
Springer Science & Business Media, 17 de gen. 2005 - 114 pàgines
In recent years, geometry has played a lesser role in undergraduate courses than it has ever done. Nevertheless, it still plays a leading role in mathematics at a higher level. Its central role in the history of mathematics has never been disputed. It is important, therefore, to introduce some geometry into university syllabuses. There are several ways of doing this, it can be incorporated into existing courses that are primarily devoted to other topics, it can be taught at a first year level or it can be taught in higher level courses devoted to differential geometry or to more classical topics. These notes are intended to fill a rather obvious gap in the literature. It treats the classical topics of Euclidean, projective and hyperbolic geometry but uses the material commonly taught to undergraduates: linear algebra, group theory, metric spaces and complex analysis. The notes are based on a course whose aim was two fold, firstly, to introduce the students to some geometry and secondly to deepen their understanding of topics that they have already met. What is required from the earlier material is a familiarity with the main ideas, specific topics that are used are usually redone.
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absolute conic affine map affine subspace Algebra angles Beltrami model centre circle collinear conjugate consider Corollary corresponding cosh cross-ratio cube Deduce defined denote diagonal dihedral group dimensional direct isometries disc dodecahedron edges elements elliptic plane equation faces finite fixed point formula given glide Hence homeomorphic homogeneous homogeneous co-ordinates hyperbolic geometry hyperbolic line hyperplane I(Rn identity independent points inversion isometry f isomorphic labelled lattice Lemma line at infinity lines in R2 lntroduction map f Mathematical matrix meet metric space Mobius band n x n matrices non-singular non-singular conic normal subgroup orbits orthogonal pairs parallel parallel axiom permutations perpendicular Platonic solid point at infinity polar line pole projective lines projective space Proposition prove quadratic forms quaternions rotation S1 x R2 sinh solid torus subset symmetry group tangent Theorem topology torus translation unique vertex vertices