000 04371nam a22004935i 4500
001 978-3-031-79549-7
003 DE-He213
005 20240730164054.0
007 cr nn 008mamaa
008 220601s2010 sz | s |||| 0|eng d
020 _a9783031795497
_9978-3-031-79549-7
024 7 _a10.1007/978-3-031-79549-7
_2doi
050 4 _aQA1-939
072 7 _aPB
_2bicssc
072 7 _aMAT000000
_2bisacsh
072 7 _aPB
_2thema
082 0 4 _a510
_223
100 1 _aGoldman, Ron.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
_981948
245 1 0 _aRethinking Quaternions
_h[electronic resource] /
_cby Ron Goldman.
250 _a1st ed. 2010.
264 1 _aCham :
_bSpringer International Publishing :
_bImprint: Springer,
_c2010.
300 _aXVIII, 157 p.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aSynthesis Lectures on Computer Graphics and Animation,
_x1933-9003
505 0 _aPreface -- Theory -- Computation -- Rethinking Quaternions and Clif ford Algebras -- References -- Further Reading -- Author Biography.
520 _aQuaternion 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. The goals of this monograph are to provide a fresh, geometric interpretation for quaternions, appropriate for contemporary computer graphics, based on mass-points; to present better ways to visualize quaternions, and the effect of quaternion multiplication on points and vectors in three dimensions using insights from the algebra and geometry of multiplication in the complex plane; to derive the formula for quaternion multiplication from first principles; to develop simple, intuitive proofs of the sandwiching formulas for rotation and reflection; to show how to apply sandwiching to compute perspective projections. In addition to these theoretical issues, we also address some computational questions. We develop straightforward formulas for converting back and forth between quaternion and matrix representations for rotations, reflections, and perspective projections, and we discuss the relative advantages and disadvantages of the quaternion and matrix representations for these transformations. Moreover, we show how to avoid distortions due to floating point computations with rotations by using unit quaternions to represent rotations. We also derive the formula for spherical linear interpolation, and we explain how to apply this formula to interpolatebetween two rotations for key frame animation. Finally, we explain the role of quaternions in low-dimensional Clifford algebras, and we show how to apply the Clifford algebra for R3 to model rotations, reflections, and perspective projections. To help the reader understand the concepts and formulas presented here, we have incorporated many exercises in order to clarify and elaborate some of the key points in the text. Table of Contents: Preface / Theory / Computation / Rethinking Quaternions and Clif ford Algebras / References / Further Reading / Author Biography.
650 0 _aMathematics.
_911584
650 0 _aImage processing
_xDigital techniques.
_94145
650 0 _aComputer vision.
_981949
650 1 4 _aMathematics.
_911584
650 2 4 _aComputer Imaging, Vision, Pattern Recognition and Graphics.
_931569
710 2 _aSpringerLink (Online service)
_981950
773 0 _tSpringer Nature eBook
776 0 8 _iPrinted edition:
_z9783031795480
776 0 8 _iPrinted edition:
_z9783031795503
830 0 _aSynthesis Lectures on Computer Graphics and Animation,
_x1933-9003
_981951
856 4 0 _uhttps://doi.org/10.1007/978-3-031-79549-7
912 _aZDB-2-SXSC
942 _cEBK
999 _c85274
_d85274