說明 
1 electronic text (xviii, 157 p. : ill.) : digital file 
系列 
Synthesis lectures on computer graphics and animation, 19339003 ; # 13


Synthesis digital library of engineering and computer science


Synthesis lectures on computer graphics and animation, 19339003 ; # 13

附註 
Part of: Synthesis digital library of engineering and computer science 

Series from website 

Includes bibliographical references (p. 153155) 

Preface  I. Theory  1. Complex numbers  2. A brief history of number systems and multiplication  Multiplication in dimensions greater than two  3. Modeling quaternions  Masspoints: a classical model for contemporary computer graphics  Arrows in four dimensions  Mutually orthogonal planes in four dimensions  4. The algebra of quaternion multiplication  5. The geometry of quaternion multiplication  6. Affine, semiaffine, and projective transformations in three dimensions  Rotation  Mirror image  Perspective projection  Perspective projection and singular 4 x 4 matrices  Perspective projection by sandwiching with quaternions  Rotorperspectives and rotoreflections  7. Recapitulation: insights and results  

II. Computation  8. Matrix representations for rotations, reflections, and perspective projections  Matrix representations for quaternion multiplication  Matrix representations for rotations  Matrix representations for mirror images  Matrix representations for perspective projections  9. Applications  Efficiency: quaternions versus matrices  Avoiding distortion by renormalization  Key frame animation and spherical linear interpolation  10. Summary: formulas from quaternion algebra  

III. Rethinking quaternions and Clifford algebras  11. Goals and motivation  12. Clifford algebras and quaternions  13. Clifford algebra for the plane  14. The standard model of the Clifford algebra for three dimensions  Scalars, vectors, bivectors, and pseudoscalars  Wedge product and cross product  Duality  Bivectors  Quaternions  15. Operands and operators: masspoints and quaternions  Odd order: masspoints  Even order: quaternions  16. Decomposing masspoints into two mutually orthogonal planes  Action of q(b, [theta]), on b  Action of q(b, [theta]), on b  Sandwiching  17. Rotation, reflection, and perspective projection  Rotation  Mirror image  Perspective projection  18. Summary  19. Some simple alternative homogeneous models for computer graphics  

References  Further reading  Author biography 

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Quaternion multiplication can be used to rotate vectors in threedimensions. Therefore, in computer graphics, quaternions have three principal applications: to increase speed and reduce storage for calculations involving rotations, to avoid distortions arising from numerical inaccuracies caused by floating point computations with rotations, and to interpolate between two rotations for key frame animation. Yet while the formal algebra of quaternions is wellknown in the graphics community, the derivations of the formulas for this algebra and the geometric principles underlying this algebra are not well understood 

Also available in print 

MorganIISLIB 
主題 
Quaternions


Complex number


Masspoint


Rotation


Reflection


Perspective projection


Quaternion


Sandwiching

