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# Topic: Quaternions and spatial rotation

 Quaternion Quaternions are sometimes used in computer graphics (and associated geometric analysis) to represent rotations or orientations of objects in 3d space. The conjugate of a quaternion corresponds to the conjugate transpose of the matrix. In this representation, the conjugate of a quaternion corresponds to the transpose of the matrix. www.ebroadcast.com.au /lookup/encyclopedia/qu/Quaternian.html   (1193 words)

 Quarternions Encyclopedia Article @ PSAMathe.net (PSA Mathe)   (Site not responding. Last check: 2007-10-09) The algebra of quaternions is often denoted by H (for Hamilton), or in flboard bold by Quaternions are often used in computer graphics (and associated geometric analysis) to represent rotations (see quaternions and spatial rotation) and orientations of objects in three-dimensional space. Using quaternions also reduces overhead from that when rotation matrices are used, because one carries only four components, not nine, the multiplication algorithms to combine successive rotations are faster, and it is far easier to renormalize the result afterwards. www.psamathe.net /encyclopedia/Quarternions   (3560 words)

 coordinate_rotation   (Site not responding. Last check: 2007-10-09) In linear algebra and geometry, a rotation is a type of transformation from one system of coordinates to another system of coordinates such that distance between any two points remains invariant under the transformation. In ordinary three dimensional space, a coordinate rotation can be defined by three Euler angles, or by a single angle of rotation and the direction of a vector about which to rotate. Rotations about the origin are most easily calculated using a 3 by 3 matrix transformation called a rotation matrix. www.lasvegaspokerdealers.com /wiki/?title=Coordinate_rotation   (585 words)

 Quaternions and spatial rotation - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-10-09) The algebra of quaternions is a useful mathematical tool for formulating the composition of arbitrary spatial rotations, and establishing the correctness of algorithms founded upon such compositions. The composition of two rotations corresponds to quaternion multiplication: if the rotation f is represented by conjugation with the quaternion z and the rotation g is represented by conjugation with w, then the composition f ∘ g is represented by conjugation with zw. The angle between two quaternions should not be confused with the angle of rotation involved in the rotation between the orientations corresponding to these quaternions: the former is half of the latter (or 180° minus half the latter). en.wikipedia.org /wiki/Quaternions_and_spatial_rotation   (6499 words)

 More on Quaternions Under this multiplication, the basis quaternions, with their negatives, form the quaternion group of order 8, Q8.The scalar part of the quaternion is a while the remainder is the vector part.Thus a vector in the context of quaternions has zero for scalar part. As is explained in more detail in quaternions and spatial rotation, the multiplicative group of non-zero quaternions acts by conjugation on the copy of R3 consisting of quaternions with real part equal to zero. Quaternion plaque on Brougham (Broom) Bridge,, which says: Here as he walked by on the 16th of October 1843 Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication i2 = j2 = k2 = i j k = −1 and cut it on a stone of this bridge. www.artilifes.com /quaternions.htm   (3319 words)

 Wikinfo | Quaternion   (Site not responding. Last check: 2007-10-09) The quaternions are an extension of the real numbers, similar to the complex numbers, except: they have dimension 4 rather than 2 over the real numbers, and the multiplication of quaternions is not commutative. Quaternions were discovered by William Rowan Hamilton of Ireland in 1843. While this is debatable in three dimensions, quaternions cannot be used in other dimensions (though extensions like octonions and Clifford algebras may be more applicable). www.wikinfo.org /wiki.php?title=Quaternion   (1362 words)

 Quaternion consisting of quaternions with real part equal to zero: it is not hard to see that the conjugation by a unit quaternion of real part cos t is a rotation by an angle 2t, the axis of the rotation being the direction of the imaginary part. Moreover, this representation of a quaternion corresponding to a spatial rotation is equivalent to the rotation operator for If two quaternions with zero scalar parts are multiplied, the scalar part of the product is the dot product of the vector parts, while the vector part of the product is the cross product. www1.gantep.edu.tr /~olgar/Quaternion.htm   (845 words)

 Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace and Virtual Reality by Princeton ... Quaternions are a delight, are very interesting to work with, and are suprisingly productive in use. Quaternions are a bit like complex numbers, but with three different complex parts instead of one. The book is primarily an exposition of the quaternion, a 4-tuple, and its primary application in a rotation operator. www.negative-procreative.biz /stuff-0691102988.html   (1809 words)

 On Quaternions   (Site not responding. Last check: 2007-10-09) A quaternion may thus be said to consist generally of a real part and a vector. And it has since appeared to Sir William Hamilton that the subject of quaternions may be illustrated by considering more closely, though briefly, this question of the determination of a fourth proportional to three rectangular directions in space, rather in a geometrical than in an algebraical point of view. And he conceives that two examples in particular, one relating to the composition of translations, and the other to the composition of rotations in space, may usefully be selected for statement on the present occasion. www.maths.tcd.ie /pub/HistMath/People/Hamilton/Quatern2/Quatern2.html   (1750 words)

 Random Works of the Web » Blog Archive » Quaternion   (Site not responding. Last check: 2007-10-09) = −1, the quaternions are obtained by adding the elements i, j and k to the real numbers which satisfy the following relations. Hamilton also described a quaternion as an ordered quadruple (4-tuple) of real numbers, and described the first coordinate as the ’scalar’ part, and the remaining three as the ‘vector’; part. Using quaternions also reduces overhead from that when rotation matrices are used, because one carries only four components, not nine, and the multiplication algorithms to combine successive rotations are faster. random.dragonslife.org /quaternion/1993   (3015 words)

 Acupressure Charts -- Recommendations and Resources   (Site not responding. Last check: 2007-10-09) In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold. This explains why, for example, the Euler angles appear to give a variable in the 3-torus, and the quaternions in a 3-sphere. gimbal lock), while the quaternion representation is always a double cover, with ''q'' and −''q'' giving the same rotation. www.becomingapediatrician.com /health/1/acupressure-charts.html   (1000 words)

 Home > Bayport, New York, NY, 11705, Bayport Real Estate, Bayport Yellow Pages, Bayport Classifieds, Bayport News, ...   (Site not responding. Last check: 2007-10-09) where r is a unit vector of H, commutative subrings of the quaternion ring. The Euclidean outer-product is not used often; however, it is mentioned as a pair with the inner-product: Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clark Maxwell. www.bayportnyus.com /topic/Quaternions   (3364 words)

 FAQTs - Knowledge Base - View Entry - Physics: Dynamics: 3D: Link: Overview: Can you give an overview of links? http://www.faqts.com/knowledge_base/view.phtml/aid/39299/fid/1856 --- Physics: Dynamics: 3D: Rotation: Angle: Euler: History: Who introduced the term 'Euler angle'? http://www.faqts.com/knowledge_base/view.phtml/aid/39939/fid/1857 --- Physics: Dynamics: 3D: Rotation: To which axes do the Euler rotation equations refer to? [principal] http://www.faqts.com/knowledge_base/view.phtml/aid/39357/fid/1857 --- Stability of free rotation http://theory.ph.man.ac.uk/~mikeb/lecture/pc167/rigidbody/stability.htm l --- Physics: Dynamics: 3D: Poinsot construction: Link: Overview: Can you give an overview of links? www.faqts.com /knowledge_base/view.phtml/aid/39259   (219 words)

 CS5984 Algorithms in Structural Bioinformatics   (Site not responding. Last check: 2007-10-09) Course@Stanford: Algorithms for Structure and Motion in Biology Torsion Angle, 3D Rotation: definition, rotation matrix, angle-axis representation, Euler angles Closed-Form Solution of Absolute Orientation using Unit Quaternions, Journal of the Optical Society A, Vol. people.cs.vt.edu /~vchoi/5984   (157 words)

 physics - Simple Lie group   (Site not responding. Last check: 2007-10-09) The Dynkin diagram has two nodes that are not connected. There is a surjective homomorphism from SO(3)* × SO(3)* to SO(4) given by quaternion multiplication; see quaternions and spatial rotation. Therefore the simple groups here start with D physics.usc.edu /~bars/symmetries/SimpleLieAlgebra.htm   (465 words)

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