## Geometry: A Comprehensive Course"A lucid and masterly survey." — Mathematics Gazette Professor Pedoe is widely known as a fine teacher and a fine geometer. His abilities in both areas are clearly evident in this self-contained, well-written, and lucid introduction to the scope and methods of elementary geometry. It covers the geometry usually included in undergraduate courses in mathematics, except for the theory of convex sets. Based on a course given by the author for several years at the University of Minnesota, the main purpose of the book is to increase geometrical, and therefore mathematical, understanding and to help students enjoy geometry. Among the topics discussed: the use of vectors and their products in work on Desargues' and Pappus' theorem and the nine-point circle; circles and coaxal systems; the representation of circles by points in three dimensions; mappings of the Euclidean plane, similitudes, isometries, mappings of the inversive plane, and Moebius transformations; projective geometry of the plane, space, and n dimensions; the projective generation of conics and quadrics; Moebius tetrahedra; the tetrahedral complex; the twisted cubic curve; the cubic surface; oriented circles; and introduction to algebraic geometry. In addition, three appendices deal with Euclidean definitions, postulates, and propositions; the Grassmann-Pluecker coordinates of lines in S3, and the group of circular transformations. Among the outstanding features of this book are its many worked examples and over 500 exercises to test geometrical understanding. |

### Contents

PRELIMINARY NOTIONS | 1 |

VECTORS | 20 |

CIRCLES | 56 |

COAXAL SYSTEMS OF CIRCLES | 106 |

THE REPRESENTATION OF CIRCLES BY POINTS IN SPACE OF THREE DIMENSIONS | 126 |

MAPPINGS OF THE EUCLIDEAN PLANE | 160 |

MAPPINGS OF THE INVERSIVE PLANE | 206 |

THE PROJECTIVE PLANE AND PROJECTIVE SPACE | 242 |

THE PROJECTIVE GEOMETRY OF n DIMENSIONS | 271 |

THE PROJECTIVE GENERATION OF CONICS AND QUADRICS | 321 |

PRELUDE TO ALGEBRAIC GEOMETRY | 401 |

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### Common terms and phrases

affine algebraic angle axes C₁ called coaxal system collineation complex numbers conic consider contains corresponding cross-ratio cubic curve deduce defined Desargues Theorem dimensions direct isometry distinct points equal equation equivalent Euclidean plane Exercise fixed points four points Gauss plane given circles given line given point harmonic conjugates Hence homogeneous coordinates hyperplane infinity inversive plane lies line at infinity line reflexion linear linearly independent locus M-transformation mapping matrix midpoints Moebius transformation nine-point circle non-singular obtain orthogonal p-line p-triangle pass pencils of planes perpendicular plane section point of intersection point-circle points X1 polar plane projective plane projective transformation proof prove quadric surface radical axis radius real Euclidean plane real numbers regulus represents rotation set of points Show sides similitude skew lines space subgroup subspaces tangent plane touch triangle ABC unique vectors vertex vertices X₁ zero