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	NationMaster - Encyclopedia: Hyperbolic quaternion
		A hyperbolic quaternion is a mathematical concept first suggested by Alexander MacFarlane in 1891 in a speech to the American Association for the Advancement of Science.
		The conjugate of a quaternion corresponds to the conjugate 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.
	www.nationmaster.com /encyclopedia/Hyperbolic-quaternion (1907 words)

	Random Works of the Web » Blog Archive » Quaternion (Site not responding. Last check: 2007-09-18)
		The algebra of quaternions is often denoted by H (for Hamilton), or in flboard bold by.
		The conjugation by a unit quaternion (a quaternion of absolute value 1) with real part cos(t) is a rotation by an angle 2t, the axis of the rotation being the direction of the imaginary part.
		The division of a quaternion by a scalar is equivalent to multiplication by the scalar inverse, such that each element of the quaternion is divided by the divisor.
	random.dragonslife.org /quaternion/1993 (3015 words)

	Quaternion (Site not responding. Last check: 2007-09-18)
		The quaternions are an example of a division ring, an algebraic structure similar to a field except for commutativity of multiplication.
		Quaternions form a 4-dimensional associative algebra over the reals (in fact a division algebra) and contain the complex numbers, but they do not form an associative algebra over the complex numbers.
		Quaternions also see use in control theory, signal processing, attitude control, physics, bioinformatics (see: Root mean square deviation (bioinformatics)), and orbital mechanics, mainly for representing rotations/orientations in three dimensions.
	www.1bx.com /en/Quaternion_physics.htm (3760 words)

	Class quaternion Documentation (Site not responding. Last check: 2007-09-18)
		Quaternions are in fact part of a small hierarchy of structures built upon the real numbers, which comprise only the set of real numbers (traditionally named
		An addition and a multiplication is defined on the set of quaternions, which generalize their real and complex counterparts.
		Quaternions (and their kin) are described in far more details in this other document (with errata and addenda).
	www.boost.org /libs/math/quaternion/quaternion.html (2345 words)

	Quaternion Summary
		Thus, a quaternion is a "number" of the form a + bi + cj + dk, where i, j, and k are the three imaginary units.
		A quaternion of unit length, q = a + bi + cj + dk, represents a rotation in the following way: The real part, a, is the cosine of half the angle of rotation, while the imaginary part, bi + cj + dk, is a vector that points along the axis of rotation.
		Quaternions also see use in control theory, signal processing, attitude control, physics, and orbital mechanics, mainly for representing rotations/orientations in three dimensions.
	www.bookrags.com /Quaternion (4151 words)

	Vector exponential
		The exponential of a quaternion is not much more complicated, since any scalar which could be added would commute with the quaternion, so its exponential could just be set aside as a multiplying scalar factor.
		Figure: Arc length along the rectangular hyperbola defines hyperbolic trigonometry when the Lorentz metric is used.
		Figure: Quaternion exponentials and sequences of quaternion exponentials can be visualized in a nomogram based on the unit one-sheeted hyperboloid of radius squared -1.
	delta.cs.cinvestav.mx /~mcintosh/comun/summer99/mcintosh/node46.html (167 words)

	Reference.com/Encyclopedia/Musean hypernumber
		Circular quaternions and octonions from the Musean hypernumbers are identical to quaternions and octonions from Cayley-Dickson construction.
		MacFarlane's hyperbolic quaternions (first mention in 1891), which are not associative.
		Conic quaternions are isomorphic to tessarines, and also to bicomplex numbers (from the multicomplex numbers).
	www.reference.com /browse/wiki/Conic_quaternion (1235 words)

	POV-Ray: Documentation: 2.4.1.6 Julia Fractal
		Quaternions have non-commutative multiplication and hypercomplex numbers can fail to have a multiplicative inverse for some non-zero elements (it has been proved that you cannot successfully generalize complex numbers to four dimensions with all the field properties intact, so something has to break).
		Of the two, the quaternions are much better known, but one can argue that hypercomplex numbers are more useful for our purposes, since complex valued functions such as sin, cos, etc. can be generalized to work for hypercomplex numbers in a uniform way.
		The first renderings of julia fractals using quaternions were done by Alan Norton and later by John Hart in the '80's.
	www.povray.org /documentation/view/3.6.1/280 (812 words)

	More on Quaternions
		By linearity, multiplication of quaternions is completely determined by the multiplication table for the basis quaternions; this table is given at the right.
		By using the distance function d(z, w) = z − w, the quaternions form a metric space (isometric to the usual Euclidean metric on R4) and the arithmetic operations are continuous.
		Quaternions were introduced by William Rowan Hamilton of Ireland in 1843.
	www.artilifes.com /quaternions.htm (3319 words)

	Unifying Gravity, Electricity and Magnetism
		Simple quaternion equations with unfamiliar collections of signs will be examined closely to see is the standard structure is there, along with a few surprises.
		In the matrix representation of a quaternion, the scalar forms the diagonal, so transposing the matrix leaves the diagonal unchanged, so it is symmetric.
		Since we have two equations, one elliptic, the other hyperbolic, that contain all the terms of the Maxwell equations, it would be a shame not to recreate the explicit form of the Maxwell equations, given their enormous success.
	world.std.com /~sweetser/quaternions/gravity/unified/unified.html (2402 words)

	The Dispatch - Serving the Lexington, NC - News (Site not responding. Last check: 2007-09-18)
		Increasing dimensionality introduces algebraic complications: Quaternion multiplication is not commutative anymore, octonion multiplication additionally is non-associative, and sedenions do not form a normed space with multiplicative norm.
		Because quaternions and octonions offer a (multiplicative) norm similar to lengths in four and eight dimensional Euclidean vector space respectively, these numbers can be referred to as points in some higher-dimensional Euclidean space.
		A special case are the bicomplex numbers which are isomorphic to tessarines, conic quaternions (from MusÃ¨s' hypernumbers), and are also contained in the 'hypercomplex number' definition by Kantor and Solodovnikov.
	www.the-dispatch.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Hypercomplex_number (798 words)

	Hyperbolic quaternion Information
		Recall that the scalar part of quaternion q = a+bi+cj+dk is the variable a.
		In particular, the hyperboloid {q: a > 0, qq* = 1 } presents a kinematic model since (with appropriate units for a,b,c, and d) it represents the locus of temporal potential for a particle passing throught the origin after a moment of local time.
		Then relative to u, arbitrary hyperbolic quaternions p and q represent simultaneous events in Minkowski space if the scalar part of the product (p - q)u* is zero.Clearly simultaneity is a function of rapidity a and direction r.Geometrically, the hyperbolic quaternions p - q and u are hyperbolic-orthogonal.
	www.bookrags.com /wiki/Hyperbolic_quaternion (949 words)

	tools.nb
		The arcsine of q equals minus V normalized to V times the hyperbolic arcsine of q times V normalized to V. The arccosine of q equals minus V normalized to V times the hyperbolic arccosine of q.
		The hyperbolic arccosine of q equals the natural log of (q plus or minus the square root of q squared minus one).
		The hyperbolic arctangent of q equals one half times the natural log of (one plus q over one minus q).
	world.std.com /~sweetser/quaternions/intro/tools/s.html (981 words)

	Quaternions
		Explanation in the historiography of mathematics: the case of Hamilton's quaternions.
		Quaternions and the heuristic role of mathematical structures in physics.
		From the universal arithmetic of Newton to the quaternion algebra of Hamilton.
	math.fullerton.edu /mathews/c2003/QuaternionBib/Links/QuaternionBib_lnk_3.html (1094 words)

	The Dispatch - Serving the Lexington, NC - News (Site not responding. Last check: 2007-09-18)
		Thus every quaternion is uniquely expressible in the form a + b i + c j + d k where a, b, c, and d are real numbers.
		While H is a four-dimensional vector space, one speaks of the scalar part of the quaternion as being a, while the vector part is the remainder b i + c j + d k.
		[1] The mathematical quaternion partakes of both these elements; in technical language it may be said to be "time plus space", or "space plus time": and in this sense it has, or at least involves a reference to, four dimensions.
	www.the-dispatch.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=quaternion (3861 words)

	Amazon.com: "hyperbolic triangle": Key Phrase page (Site not responding. Last check: 2007-09-18)
		whereas the sum of the angles of a hyperbolic triangle is always less than 180.
		hyperbolic triangle with vertices on the horocircle {z E IHI I Ini(z) = 11.
		Hyperbolic Geometry 13 The Gauss-Bonnet formula shows that the hyperbolic area of a hyperbolic triangle depends only on its angles.
	www.amazon.com /phrase/hyperbolic-triangle (522 words)

	Hypercomplex Electromagnetic Theory (Site not responding. Last check: 2007-09-18)
		These may not look like quaternion expressions, but they are, as we showed on the Hypercomplex Math page (use browser BACK button to come back).
		As I mentioned earlier, vector analysis was developed as a kind of shorthand for quaternion analysis, keeping all expressions to three dimensions or less.
		For quaternion formulations of electromagnetism, see Sweetser and Jack.
	home.usit.net /~cmdaven/eelectro.htm (1014 words)

	Poincare Disk
		Hyperbolic geometry and Hölder continuity of conformal mappings.
		The hyperbolic geometry with horocycles as primitive notions.
		On a new presentation of the hyperbolic trigonometry by aid of the Poincaré model.
	math.fullerton.edu /mathews/c2003/poincaredisk/PoincareDiskBib/Links/PoincareDiskBib_lnk_3.html (1005 words)

	Amazon.com: "hyperbolic area": Key Phrase page (Site not responding. Last check: 2007-09-18)
		the hyperbolic plane and the three basic laws of trigonometry in the hyperbolic plane.
		Hyperbolic area and the Gauss - Bonnet formula For a subset Ac we define (A), the hyperbolic area of A, by p(A)=J...
		The hyperbolic area of a measurable subset Sl C ID)...
	www.amazon.com /phrase/hyperbolic-area (569 words)

	A-Index
		Returns the principal inverse hyperbolic cosine of this quaternion q.
		Returns the principal inverse hyperbolic cotangent of this quaternion q.
		Returns the sum of this quaternion q and the imaginary number jx.
	cheetah.vlsi.uwaterloo.ca /~dwharder/java/doc/index-files/index-1.html (1530 words)

	MT A618 Hyperbolic 3-Manifolds (Site not responding. Last check: 2007-09-18)
		Before Thurston's work in the 1970's, examples of hyperbolic 3-manifolds were exceedingly rare; only a couple of simple constructions were known.
		Indeed, in higher dimensions this is still the case (probably because hyperbolic 4-manifolds really are pretty rare).
		The image above is a hyperbolic icosahedron which serves as a fundamental domain for the five-fold cyclic branched cover of the figure-eight knot (one of the "Fibonacci manifolds").
	borel.slu.edu /teach/618/index.html (264 words)

	S-Index
		Returns the hyperbolic secant of this complex number z.
		Returns a normalized form of this quaternion q when it is non-zero and q otherwise.
		Returns the difference between this quaternion q and the quaternion r = e + if + jc + kd.
	cheetah.vlsi.uwaterloo.ca /~dwharder/java/doc/index-files/index-17.html (450 words)

	New complex- and quaternion-hyperbolic reflection groups, Daniel Allcock
		[2] —, The Leech lattice and complex hyperbolic reflections, to appear in Invent.
		[3] D. Allcock, J. Carlson, and D. Toledo, A complex hyperbolic structure for moduli of cubic surfaces, C. Acad.
		[4] —, The complex hyperbolic geometry of the moduli space of cubic surfaces, preprint, 2000.
	projecteuclid.org /Dienst/UI/1.0/Display/euclid.dmj/1092749438 (467 words)

	Hyperbolic quaternion
		Encyclopedia : H : HY : HYP : Hyperbolic quaternion
		is a hyperbolic quaternion when a, b, c, and d are real numbers and the basis set has these products:
		The basis [\] of the vector space of hyperbolic quaternions is not closed under multiplication: for example, [ji=-k].
	encycl.opentopia.com /term/Hyperbolic_quaternion (923 words)

	preprint (Site not responding. Last check: 2007-09-18)
		Hyperbolic surfaces with prescribed infinite symmetry groups
		Hyperbolic geometry and the moduli space of real binary sextics with J. Carlson and D. Toledo
		A Complex Hyperbolic Structure for Moduli of Cubic Surfaces with James A. Carlson, Domingo Toledo
	www.mathnet.or.kr /API/?MIval=research_prof_papers_detail&u_id=Allcock (189 words)

	Hyperbolic quaternion Details, Meaning Hyperbolic quaternion Article and Explanation Guide
		Hyperbolic quaternion Guide, Meaning, Facts, Information and Description
		If q* = a - bi - cj - dk is the conjugate of q, then the product q q* = aa - bb - cc - dd is the quadratic form used in Minkowski space.
		In it he establishes a model for hyperbolic space on the hyperboloid q q* = 1.
	www.e-paranoids.com /h/hy/hyperbolic_quaternion.html (218 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: 2007-09-18)
		The way had been prepared for Minkowski's space by the development of hyperbolic quaternions in the 1890s.
		In fact, as a mathematical structure, Minkowski space can be taken as hyperbolic quaternions, minus the multiplicative product, and retaining only the bilinear form
		which is generated by the hyperbolic quaternion product pq*.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Minkowski_metric (1101 words)

	Research (Daniel Allcock)
		Hyperbolic geometry and the moduli space of real binary sextics
		The complex hyperbolic geometry of the moduli space of cubic surfaces
		A complex hyperbolic structure for moduli of cubic surfaces
	www.ma.utexas.edu /~allcock (263 words)

	Hyperbolic quaternion Did You Mean hyperbolic?
		Add a link on the top of this hyperbolic quaternion page Express submission by secure payment !
		Article on Hyperbolic quaternion, category, different spelling or sense
		Page Hyperbolic quaternion cached on Friday 16th of February 2007 08:17:26 AM Compteur gratuit
	www.did-you-mean.com /Hyperbolic_quaternion.html (428 words)

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