A Course in Modern GeometriesSpringer New York, 1989 - 232 sidor A Course in Modern Geometries is designed for a junior-senior level course for mathematics majors, including those who plan to teach in secondary school. Chapter 1 presents several finite geometries in an axiomatic framework. Chapter 2 introduces Euclid's geometry and the basic ideas of non-Euclidean geometry. The synthetic approach of Chapters 1 - 2 is followed by the analytic treatment of transformations of the Euclidean plane in Chapter 3. Chapter 4 presents plane projective geometry both synthetically and analytically. The extensive use of matrix representations of groups of transformations in Chapters 3 - 4 reinforces ideas from linear algebra and serves as excellent preparation for a course in abstract algebra. Each chapter includes a list of suggested sources for applications and/or related topics. |
Innehåll
CHAPTER | 1 |
NonEuclidean Geometry | 12 |
Gaining Perspective | 25 |
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absolute conic affine geometry algebra angle sum asymptotic triangles Axiom axiomatic system collinear points concurrent lines congruent construct contains corresponding Definition direct isometry distinct points elements elliptic geometry equivalence class Euclid's Euclidean geometry Euclidean plane exactly Exercise Figure Find the matrix finite projective planes following theorem frieze group frieze pattern glide reflection half-turns homogeneous coordinates homogeneous parameters hyperbolic geometry ideal points incident indirect invariant point line of symmetry line perpendicular Mathematics matrix representation midpoint non-Euclidean geometry one-to-one linear transformation P₁ pair pencil of points pencils of lines point conic point of intersection points and lines polar projective geometry proof of Theorem properties Prove Theorem real numbers reflection with axis respectively right angles rotation with center Saccheri quadrilateral segment sensed parallel set of points sides similar similarity geometry straight line symmetry group tangent u₁ u₂ ultraparallel vector verify