## Modern Differential Geometry of Curves and Surfaces with Mathematica, Third EditionPresenting theory while using Mathematica in a complementary way, Modern Differential Geometry of Curves and Surfaces with Mathematica, the third edition of Alfred Gray’s famous textbook, covers how to define and compute standard geometric functions using Mathematica for constructing new curves and surfaces from existing ones. Since Gray’s death, authors Abbena and Salamon have stepped in to bring the book up to date. While maintaining Gray's intuitive approach, they reorganized the material to provide a clearer division between the text and the Mathematica code and added a Mathematica notebook as an appendix to each chapter. They also address important new topics, such as quaternions. The approach of this book is at times more computational than is usual for a book on the subject. For example, Brioshi’s formula for the Gaussian curvature in terms of the first fundamental form can be too complicated for use in hand calculations, but Using the techniques described in this book, readers will understand concepts geometrically, plotting curves and surfaces on a monitor and then printing them. Containing more than 300 illustrations, the book demonstrates how to use |

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### Contents

Curves in the Plane | 5 |

Famous Plane Curves | 39 |

Alternative Ways of Plotting Curves | 73 |

New Curves from Old | 99 |

Determining a Plane Curve from its Curvature | 127 |

Determining a Plane Curve from Its Curvature 127 5 Determining a Plane Curve from Its Curvature | 128 |

Notebook 5 | 146 |

Global Properties of Plane Curves | 153 |

Notebook 14 | 452 |

Surfaces of Revolution and Constant Curvature | 461 |

Notebook 15 | 488 |

A Selection of Minimal Surfaces | 501 |

Intrinsic Surface Geometry | 531 |

Notebook 17 | 548 |

Asymptotic Curves and Geodesics on Surfaces | 557 |

Notebook 18 | 582 |

Notebook 6 | 181 |

Curves in Space | 191 |

Notebook 7 | 217 |

Construction of Space Curves | 229 |

Notebook 8 | 254 |

Calculus on Euclidean Space | 263 |

Notebook 9 | 283 |

Surfaces in Euclidean Space | 287 |

Notebook 10 | 320 |

Nonorientable Surfaces | 331 |

Notebook 11 | 352 |

Metrics on Surfaces | 361 |

Notebook 12 | 379 |

Shape and Curvature | 385 |

Notebook 13 | 420 |

Ruled Surfaces | 431 |

Principal Curves and Umbilic Points | 593 |

Canal Surfaces and Cyclides of Dupin | 639 |

The Theory of Surfaces of Constant Negative Curvature | 683 |

Notebook 21 | 712 |

Minimal Surfaces via Complex Variables | 719 |

Rotation and Animation Using Quaternions | 767 |

Differentiable Manifolds | 809 |

Riemannian Manifolds | 847 |

Notebook 25 | 868 |

Abstract Surfaces and Their Geodesics | 871 |

The GaussBonnet Theorem | 901 |

931 | |

953 | |

977 | |