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作者 Goldman, Ron, 1947-
書名 Rethinking quaternions [electronic resource] : theory and computation / Ron Goldman
出版項 San Rafael, Calif. (1537 Fourth Street, San Rafael, CA 94901 USA) : Morgan & Claypool, c2010
國際標準書號 9781608454211 (electronic bk.)
9781608454204 (pbk.)
國際標準號碼 10.2200/S00292ED1V01Y201008CGR013 doi
book jacket
說明 1 electronic text (xviii, 157 p. : ill.) : digital file
系列 Synthesis lectures on computer graphics and animation, 1933-9003 ; # 13
Synthesis digital library of engineering and computer science
Synthesis lectures on computer graphics and animation, 1933-9003 ; # 13
附註 Part of: Synthesis digital library of engineering and computer science
Series from website
Includes bibliographical references (p. 153-155)
Preface -- I. Theory -- 1. Complex numbers -- 2. A brief history of number systems and multiplication -- Multiplication in dimensions greater than two -- 3. Modeling quaternions -- Mass-points: 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, semi-affine, 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: mass-points and quaternions -- Odd order: mass-points -- Even order: quaternions -- 16. Decomposing mass-points 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
Abstract freely available; full-text restricted to subscribers or individual document purchasers
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Mode of access: World Wide Web
System requirements: Adobe Acrobat Reader
Quaternion multiplication can be used to rotate vectors in three-dimensions. 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 well-known 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
主題 Quaternions
Complex number
Perspective projection
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