Brought a hundred and sixty years in the past as an try to generalize advanced numbers to better dimensions, quaternions are actually famous as some of the most very important recommendations in sleek special effects. they give a robust approach to characterize rotations and in comparison to rotation matrices they use much less reminiscence, compose quicker, and are evidently fitted to effective interpolation of rotations. regardless of this, many practitioners have kept away from quaternions as a result arithmetic used to appreciate them, hoping that a few day a extra intuitive description could be available.
The wait is over. Andrew Hanson's new ebook is a clean standpoint on quaternions. the 1st a part of the e-book specializes in visualizing quaternions to supply the instinct essential to use them, and comprises many illustrative examples to encourage why they're important―a attractive advent to these eager to discover quaternions unencumbered by way of their mathematical points. the second one half covers the all-important complex functions, together with quaternion curves, surfaces, and volumes. ultimately, for these short of the complete tale of the math in the back of quaternions, there's a mild advent to their 4-dimensional nature and to Clifford Algebras, the all-encompassing framework for vectors and quaternions.
* Richly illustrated creation for the developer, scientist, engineer, or scholar in special effects, visualization, or leisure computing.
* Covers either non-mathematical and mathematical ways to quaternions.
* significant other site with an collection of quaternion utilities and pattern code, facts units for the book's illustrations, and Mathematica notebooks with crucial algebraic utilities.
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Additional resources for Visualizing Quaternions (The Morgan Kaufmann Series in Interactive 3D Technology)
Four. 2 Hemispheres in S3 sixty nine 70 seventy one seventy two seventy three seventy four seventy six seventy six seventy seven eighty eighty two eighty three 09 VISUALIZING LOGARITHMS AND EXPONENTIALS nine. 1 advanced Numbers nine. 2 Quaternions 87 87 ninety one V I S U A L I Z I N G I N T E R P O L AT I O N M E T H O D S 10. 1 fundamentals of Interpolation 10. 1. 1 Interpolation concerns 10. 1. 2 Gram–Schmidt Derivation of the SLERP 10. 1. three † substitute Derivation 10. 2 Quaternion Interpolation 10. three an identical 3×3 Matrix procedure ninety three ninety three ninety three ninety seven ninety nine one hundred and one 104 10 xiv eleven 12 thirteen 14 CONTENTS L O O ok I N G AT E L E M E N TA R Y Q U AT E R N I O N FRAMES eleven. 1 A unmarried Quaternion body eleven. 2 a number of remoted Frames eleven. three A Rotating body series eleven. four Synopsis a hundred and five a hundred and five 106 107 one hundred ten Q U AT E R N I O N S A N D T H E B E LT T R I C okay : CONNECTING TO THE identification 12. 1 Very attention-grabbing, yet Why? 12. 1. 1 The Intuitive solution 12. 1. 2 † The Technical resolution 12. 2 the main points 12. three Frame-sequence Visualization equipment 12. three. 1 One Rotation 12. three. 2 Rotations 12. three. three Synopsis 111 113 113 113 114 118 a hundred and twenty 121 122 Q U AT E R N I O N S A N D T H E R O L L I N G B A L L : EXPLOITING ORDER DEPENDENCE thirteen. 1 Order Dependence thirteen. 2 The Rolling Ball Controller thirteen. three Rolling Ball Quaternions thirteen. four † Commutators thirteen. five 3 levels of Freedom From 123 123 one hundred twenty five 128 one hundred thirty 131 Q U AT E R N I O N S A N D G I M B A L L O C ok : L I M I T I N G T H E AVA I L A B L E S PA C E 14. 1 suggestions process Suspension 14. 2 Mathematical Interpolation Singularities 14. three Quaternion perspective 133 133 134 134 PAR T II complex QUATERNION issues 137 CONTENTS 15 xv A LT E R N AT I V E means S O F W R I T I N G Q U AT E R N I O N S 15. 1 Hamilton’s Generalization of advanced Numbers 15. 2 Pauli Matrices 15. three different Matrix varieties 141 142 143 a hundred and forty four sixteen potency AND COMPLEXITY matters sixteen. 1 Extracting a Quaternion sixteen. 1. 1 confident hint R sixteen. 1. 2 Nonpositive hint R sixteen. 2 potency of Vector Operations 147 148 149 149 a hundred and fifty 17 A D VA N C E D S P H E R E V I S U A L I Z AT I O N 17. 1 Projective technique 17. 1. 1 The Circle S1 17. 1. 2 common SN Polar Projection 17. 2 Distance-preserving pulling down tools 17. 2. 1 Unroll-and-Flatten S1 17. 2. 2 S2 Flattened Equal-area approach 17. 2. three S3 Flattened Equal-volume process 153 153 153 one hundred fifty five 156 157 157 159 18 extra ON LOGARITHMS AND EXPONENTIALS 18. 1 2nd Rotations 18. 2 3D Rotations 18. three utilizing Logarithms for Quaternion Calculus 18. four Quaternion Interpolations as opposed to Log one hundred sixty five a hundred sixty five 167 171 171 TWO-DIMENSIONAL CURVES 19. 1 Orientation Frames for 2nd house Curves 19. 1. 1 2nd Rotation Matrices 19. 1. 2 The body Matrix in 2nd 19. 1. three body Evolution in second 19. 2 what's a Map? 19. three Tangent and basic Maps 19. four sq. Root shape 173 173 174 a hundred seventy five 176 176 177 179 19 xvi CONTENTS 19. four. 1 body Evolution in (a, b) 19. four. 2 Simplifying the body Equations 20 21 179 179 three-d CURVES 20. 1 creation to 3D house Curves 20. 2 normal Curve Framings in 3D 20. three Tubing 20. four Classical Frames 20. four. 1 Frenet–Serret body 20. four. 2 Parallel shipping body 20. four. three Geodesic Reference body 20. four. four common Frames 20. five Mapping the Curvature and Torsion 20.