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Topic: Quaternion


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  Quaternions - LoveToKnow 1911
Quaternions (as a mathematical method) is an extension, or improvement, of Cartesian geometry, in which the artifices of co-ordinate axes, andc., are got rid of, all directions in space being treated on precisely the same terms.
The evolution of quaternions belongs in part to each of two weighty branches of mathematical history - the interpretation of the imaginary (or impossible) quantity of common algebra, and the Cartesian application of algebra to geometry.
Even in Hamilton's earlier work it was shown that all such questions were reducible to the solution of linear equations in quaternions; and he proved that this, in turn, depended on the determination of a certain operator, which could be represented for purposes of calculation by a single symbol.
www.1911encyclopedia.org /Quaternions   (0 words)

  
 Quaternion Gauss Maps and Optimal Framings of Curves and Surfaces - Hanson (ResearchIndex)
Our fundamental tool is the "quaternion Gauss map," a generalization to quaternion space of the tangent map...
Quaternion gauss maps and optimal framings of curves and surfaces.
3 An Elementary Treatise on Quaternions (context) - TAIT
citeseer.ist.psu.edu /hanson98quaternion.html   (791 words)

  
  UnrealWiki: Quaternion
Quaternions are a mathematical construct that represents a rotation in three dimensions.
The rotation of an object defined by a quaternion is the facing as defined by the vector, rolled by the roll component of the quaternion.
The quaternions occupy a smooth, seamless, isotropic space which is a generalization of the surface of a sphere.
wiki.beyondunreal.com /wiki/Quaternion   (1260 words)

  
 NationMaster - Encyclopedia: Quaternion   (Site not responding. Last check: )
In this representation, the conjugate of a quaternion corresponds to the transpose of the matrix.
Quaternions are often used in computer graphics (and associated geometric analysis) to represent rotations (see quaternions and spatial rotation) and orientations of objects in 3d space.
Quaternions, first introduced by William Hamilton, can also be written in the form a + bi + cj + dk, where a, b, c, and d are real numbers and i, j, and k are imaginary numbers, which is similar to that of complex numbers.
www.nationmaster.com /encyclopedia/Quaternion   (672 words)

  
 NationMaster - Encyclopedia: Quaternion group   (Site not responding. Last check: )
The quaternion group Q may be regarded as acting on the eight nonzero elements of the 2-dimensional vector space over the finite field GF(3).
In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (a mixture of integers and half-integers is not allowed).
The quaternion group Q may be regarded as acting on the eight nonzero elements of the 2-dimensional vector space over the finite field In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements.
www.nationmaster.com /encyclopedia/Quaternion-group   (1650 words)

  
 PlanetMath: quaternion group
The quaternion group, or quaternionic group, is a noncommutative group with eight elements.
Quaternions were known to Gauss in 1819 or 1820, but he did not publicize this discovery, and quaternions weren't rediscovered until 1843, with Hamilton.
This is version 9 of quaternion group, born on 2002-04-17, modified 2006-07-07.
www.planetmath.org /encyclopedia/QuaternionGroup.html   (197 words)

  
 quaternion
Quaternions, first introduced by William Hamilton, can also be written in the form a + bi + cj + dk, where a, b, c, and d are real numbers and i, j, and k are imaginary numbers, which is similar to that of complex numbers.
For a while, quaternions were very influential: they were taught in many mathematics departments in the United States in the late 1800s, and were a mandatory topic of study at Dublin, where Hamilton ran the observatory.
Had quaternions come along later, when theoretical physicists were trying to understand patterns among subatomic particles, they may have found a place in modern science; after all, the unit quaternions form the group SU(2), which is perfect for studying spin-½ particles.
www.daviddarling.info /encyclopedia/Q/quaternion.html   (240 words)

  
 QUATERNIONS - LoveToKnow Article on QUATERNIONS   (Site not responding. Last check: )
Quaternions (as a mathematica method) is an extension, or improvement, of Cartesian geometry in which the artifices of co-ordinate axes, andc., are got rid of, a!
Even in Hamiltons earlier work it was shown that all such questions were reducible to the solution of linear equations in quaternions; and he proved that this, in turn, depended on the determination of a certain operator, which could be represented for purposes of calculation by a single symbol.
This is the basis of a method parallel throughout to the quaternion method; in the specification of rotors and motors it is independent of the origin which for these purposes the quaternion method, pure and simple, requires.
31.1911encyclopedia.org /Q/QU/QUATERNIONS.htm   (5773 words)

  
 Hurwitz quaternion - Wikipedia, the free encyclopedia
In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (a mixture of integers and half-integers is not allowed).
It can be checked that H is closed under quaternion multiplication and addition, so that it forms a subring of the ring of all quaternions H.
A Lipschitz quaternion (or Lipschitz integer) is a quaternion whose components are all integers.
en.wikipedia.org /wiki/Hurwitz_quaternion   (447 words)

  
 Maths -Quaternion transforms - Martin Baker
To perform these transforms we first convert the input vector to a quaternion by setting the imaginary pars of the quaternion to the x,y and z values of the vector, the real part of the quaternion is set to zero.
First we convert the 3D vector into a quaternion, to do this we set the imaginary pars of the quaternion to the x,y and z values of the vector, the real part of the quaternion is set to zero.
Quaternions can represent rotations, also non-normalised quaternions can represent scaling by an equal amount in all dimensions (see discussion with minorlogic in the second part of this thread).
www.euclideanspace.com /maths/algebra/realNormedAlgebra/quaternions/transforms/index.htm   (2031 words)

  
 PlanetMath: quaternion group
The quaternion group, or quaternionic group, is a noncommutative group with eight elements.
Quaternions were known to Gauss in 1819 or 1820, but he did not publicize this discovery, and quaternions weren't rediscovered until 1843, with Hamilton.
This is version 8 of quaternion group, born on 2002-04-17, modified 2005-03-18.
planetmath.org /encyclopedia/QuaternionGroup.html   (205 words)

  
 Maths - Quaternions - Martin Baker
Quaternions form an interesting algebra where each object contains 4 scalar variables (sometimes known as Euler Parameters not to be confused with Euler angles), these objects can be added and multiplied as a single unit in a similar way to the usual algebra of numbers.
When quaternions are used in this way we can think of them as being similar to axis-angle except that real part is equal to cos(angle/2) and the complex part is made up of the axis vector times sin(angle/2).
The quaternion 'i' represents a rotation of 180 degrees about the x axis, the quaternion 'j' represents a rotation of 180 degrees about the y axis, the quaternion 'k' represents a rotation of 180 degrees about the y axis.
www.euclideanspace.com /maths/algebra/realNormedAlgebra/quaternions/index.htm   (1422 words)

  
 Class quaternion Documentation   (Site not responding. Last check: )
Quaternions (and their kin) are described in far more details in this other document (with errata and addenda).
For this version of (the mathematical objects known as) quaternions, the Euclidian norm (also known as magnitude) is the square root of the Cayley norm.
There are several reasons to this: on the one hand, the equivalent of analytic continuation for quaternions ("branch cuts") remains to be investigated thoroughly (by me, at any rate...), and we wish to avoid the nonsense introduced in the standard by exponentiations of complexes by complexes (which is well defined, but not in the standard...).
www.boost.org /libs/math/quaternion/quaternion.html   (2345 words)

  
 Visualizing Quaternions: Quaternion Maps Documentation
Quaternion Maps are plots of the quaternion values corresponding to 3D orientation frames attached to a curve, surface, or volume.
Quaternions are used in this manner to represent orientations as one sweeps along a curve in 3D.
Quaternion maps are obtained by connecting a set of quaternions to form a smooth curve, surface, or volume.
www.cs.indiana.edu /~hanson/quatvis/Quaternion-Maps/index.html   (1161 words)

  
 Quaternion - CGAFaq   (Site not responding. Last check: )
Generally, the quaternion advocates seem to feel the opponents are lazy or thick-headed, and that deeper understanding of quaternions would lead to deeper appreciation.
The norm of a quaternion, N(q), is conventionally the sum of the squares of the four components.
Quaternions can be approached and applied in numerous different ways, so if you keep looking it’s likely you will find something that suits your taste and your needs.
cgafaq.info /wiki/Quaternion   (1229 words)

  
 Delphi3D - Rapid OpenGL Development
Quaternions were originally invented as an extension to complex numbers.
The norm of a quaternion is similar to that of a vector.
The second argument that is commonly given for the use of quaternions is the possibility of smoothly interpolating between two orientations.
www.delphi3d.net /articles/viewarticle.php?article=quaternions.htm   (0 words)

  
 6DoF (Quaternion) (Aerospace Blockset)
An error in the magnitude in one element of the quaternion vector is spread equally among all the elements, potentially increasing the error in the state vector.
The first input to the block is a vector containing the three applied forces, in Newtons, and the second input is a vector containing the three applied moments, in Newton meters.
The block assumes that the applied forces are acting at the center of gravity of the body, and that the mass and inertia are constant.
www.weizmann.ac.il /matlab/toolbox/aeroblks/6dofquaternion.html   (399 words)

  
 Quaternion Julia Fractals
Quaternion Julia fractals are created by the same principle as the more traditional Julia set except that it uses 4 dimensional complex numbers instead of 2 dimensional complex numbers.
A quaternion has two more complex components and might be written as q = r + a i + b j + c k where r, a, b, and c are real numbers.
is the point in quaternion space being considered and c is a constant that identifies the particular quaternion.
local.wasp.uwa.edu.au /~pbourke/fractals/quatjulia   (0 words)

  
 Quaternion Analysis
Because of the non-commutativity of quaternions, the meaning of quaternion power series is unclear, and the most obvious generalizations of the difference quotient are useless.
This is a quaternion parallel to a, b and c whose length is the area of the 3-parallelopiped determined by a, b and c.
The left- and right-regularity of a quaternion function mean that locally, the function has the effect of multiplying on the left or right, respectively, by a quaternion (the derivative).
www.zipcon.net /~swhite/docs/math/quaternions/analysis.html   (1395 words)

  
 quaternion [jME Wiki]
Additional benefits of the Quaternion is reducing the chance of Gimbal Lock and allowing for easily interpolation between two rotations (spherical linear interpolation or slerp).
While Quaternions are quite difficult to fully understand, there are an exceeding number of convience methods to allow you to use them without having to understand the math behind it.
That is, if you have an initial Quaternion representing the original orientation of an object, and you have a final Quaternion representing the orientation you want the object to face, you can do this very smoothly with slerp.
www.jmonkeyengine.com /wiki/doku.php?id=quaternion   (679 words)

  
 HAKMEM QUATERNIONS -- DRAFT, NOT YET PROOFED
F(Q) restricted to the unit quaternions is a rigid rotation of the 3-sphere, and because this mapping is a group translation, it has no fixed point.
This shows that in the quaternion representation of the rotation group, the Haar measure is the Lebesgue measure on the 3-sphere.
Quaternions form a convenient 4-parameter representation of rotations, since composition of rotations is done by quaternion multiplication.
www.inwap.com /pdp10/hbaker/hakmem/quaternions.html   (633 words)

  
 Quaternion Julia Fractals
Quaternion Julia fractals are created by the same principle as the more traditional Julia set except that it uses 4 dimensional complex numbers instead of 2 dimensional complex numbers.
A quaternion has two more complex components and might be written as q = r + a i + b j + c k where r, a, b, and c are real numbers.
is the point in quaternion space being considered and c is a constant that identifies the particular quaternion.
astronomy.swin.edu.au /~pbourke/fractals/quatjulia   (792 words)

  
 Quaternion - lslWiki
To set the elements of the rotation type directly, compute the quaternion as follows.
To represent a rotation of a given angle around a given axis, first, convert the axis to a unit vector (a vector of magnitude 1).
element of the quaternion is equal to the cosine of half the angle.
rpgstats.com /wiki/index.php?title=Quaternions   (373 words)

  
 Class quicktime.qd3d.math.Quaternion   (Site not responding. Last check: )
Return a a quaternion that is a fast interpolation between the this and the specified quaternion.
Return a a quaternion that is a linear interpolation between the this and the specified quaternion.
Returns a quaternion that is either identical to the quaternion specified by the q1 parameter or is the negative of q1, depending on whether q1 or its negative matches the orientation of the quaternion specified by the q2 parameter.
www.brook.edu /es/dynamics/models/temp/QTJava_SDK/qtjavadocs/docs/quicktime/qd3d/math/Quaternion.html   (697 words)

  
 Ogre Forums :: View topic - Converting Quaternion to Vector
A quaternion is a rotation transformation, and a vector defines a straight track between two points, from a start to a destination.
But with a quaternion, you say "face 45 degrees clockwise from whatever direction you are already facing" (very simplified example).
Since all objects are assumed to start facing unit z then are rotated by the quaternion, taking a unit z vector and rotating it by the quaternion gives us the vector in the facing direction of the object.
www.ogre3d.org /phpBB2/viewtopic.php?p=192956   (0 words)

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