## Elementary GeometryThe geometry of two and three dimensional space has long been studied for its own sake, but its results also underlie modern developments in fields as diverse as linear algebra, quantum physics, and number theory. This text is a careful introduction to Euclidean geometry that emphasizes its connections with other subjects. Glimpses of more advanced topics in pure mathematics are balanced by a straightforward treatment of the geometry needed for mechanics and classical applied mathematics. The exposition is based on vector methods; an introductory chapter relates these methods to the more classical axiomatic approach. The text is suitable for undergraduate courses in geometry and will be useful supplementary reading for students of mechanics and mathematical methods. |

### Contents

1 | 11 |

5 | 22 |

3 | 48 |

Euclidean geometry | 63 |

Coordinates and equations | 85 |

3 | 92 |

4 | 98 |

Plane geometry | 104 |

8 | 166 |

Area and volume | 192 |

Quadric surfaces | 223 |

Differential geometry of curves | 242 |

Differential geometry of surfaces | 266 |

Appendix A The trigonometric functions | 291 |

300 | |

303 | |

### Common terms and phrases

algebra arc length axis calculate called Cartesian Cavalieri's principle centre Chapter circle collinear complex numbers conic constant coordinate system corresponding defined definition denote derivative differential dimensions distance dot product eigenvalue eigenvectors ellipse equal equation Euclidean plane Euclidean space Example fact Figure formula function fundamental form Gaussian curvature geodesic geometry given hyperbola hyperboloid inequality integral isometry linear Möbius transformation multiple nonzero obtained oriented angle origin orthogonal matrix orthonormal basis P₁ parabola parallel parallelogram parameter values parameterized by arc patch of surface perpendicular piecewise regular points of intersection polar coordinates polygonal region position vector Proof Proposition prove quadric quaternion radius rational real numbers regular curve root rotation scalar triple product sides sphere straight line subset Suppose tangent line tangent vector theorem torsion unit vector vector field x'Mx zero