## Vector Analysis for Computer GraphicsIn my last book, Geometry for Computer Graphics, I employed a mixture of algebra and vector analysis to prove many of the equations used in computer graphics. At the time, I did not make any distinction between the two methodologies, but slowly it dawned upon me that I had had to discover, for the first time, how to use vector analysis and associated strategies for solving geometric problems. I suppose that mathematicians are taught this as part of their formal mathematical training, but then, I am not a mathematician! After some deliberation, I decided to write a book that would introduce the beginner to the world of vectors and their application to the geometric problems encountered in computer graphics. I accepted the fact that there would be some duplication of formulas between this and my last book; however, this time I would concentrate on explaining how problems are solved. The book contains eleven chapters: The first chapter distinguishes between scalar and vector quantities, which is reasonably straightforward. The second chapter introduces vector repres- tation, starting with Cartesian coordinates and concluding with the role of direction cosines in changes in axial systems. The third chapter explores how the line equation has a natural vector interpretation and how vector analysis is used to resolve a variety of line-related, geometric problems. Chapter 4 repeats Chapter 3 in the context of the plane. |

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### Contents

II | 1 |

III | 7 |

IV | 11 |

V | 13 |

VII | 14 |

VIII | 17 |

IX | 18 |

X | 19 |

XLV | 130 |

XLVI | 133 |

XLVII | 138 |

XLVIII | 140 |

XLIX | 142 |

L | 144 |

LI | 146 |

LII | 149 |

XI | 22 |

XII | 36 |

XIII | 44 |

XIV | 48 |

XV | 52 |

XVI | 54 |

XVII | 57 |

XVIII | 61 |

XIX | 65 |

XX | 68 |

XXI | 69 |

XXII | 74 |

XXIII | 77 |

XXIV | 81 |

XXV | 86 |

XXVI | 89 |

XXVII | 93 |

XXVIII | 94 |

XXX | 97 |

XXXI | 101 |

XXXII | 103 |

XXXIII | 104 |

XXXIV | 106 |

XXXV | 108 |

XXXVI | 110 |

XXXVII | 113 |

XXXVIII | 114 |

XXXIX | 115 |

XL | 118 |

XLI | 120 |

XLII | 123 |

XLIII | 126 |

XLIV | 129 |

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### Common terms and phrases

a x b x c a+bi algebra axis bump map calculating Cartesian form Commutative law complex number components computer graphics coordinates cross product cylinder defined direction cosines discover dot product equals example Figure geometric given gives interpolated light source line equation line intersecting line segments located magnitude multiply Eq normal vector object obtain origin parallel parametric form polygon position vector problem projection plane quaternion qv¯q reference to Fig reflected represented rotated scalar product shown in Fig shows sphere Substituting Eq surface normal tangential test Eq transform triangle triple product unit vector vector analysis vector product X Y Z xaxb xaxc xbya Xv Yv zaxb zazb zazc zero zmin