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PlanetMath: bivector A bivector is a two-dimensional analog to a one-dimensional vector. Whereas a vector is often utilized to represent a one-dimensional directed quantity (often visualized geometrically as a directed line-segment), a bivector is used to represent a two-dimensional directed quantity (often visualized as an oriented plane). Typically the orientation of the bivector is established by placing the two vectors tail-to-tail and sweeping from the first vector to the second. planetmath.org /encyclopedia/Bivector.html   (137 words)

Springer Online Reference Works , the coordinates of a bivector behave as coordinates of a twice-contravariant tensor. The scalar product of two bivectors is the number equal to the product of the measures of the factors by the cosine of the angle between their two carrier planes. In tensor calculus a bivector is an arbitrary contravariant skew-symmetric tensor of valency 2 (i.e. eom.springer.de /B/b016600.htm   (425 words)

Outer product Geometrically a bivector x∧y is the sweeping surface generated when the vector x slips along y in the direction of y. Similarly, the product of a bivector with a third LI vector gives rise to an oriented volume, generated by sliding the bivector "area" along of the third vector. A bivector can be used to unambiguously represent a plane embedded in any n-dimensional space, while the use of the normal vector is only useful in a 3D space. www.ebroadcast.com.au /lookup/encyclopedia/ou/Outer_product.html   (409 words)

Springer Online Reference Works   (Site not responding. Last check: 2007-11-03) the bivector space may be metrized with the aid of the metric tensor Bivector spaces are used in Riemannian geometry and in the general theory of relativity. Essentially, a bivector space is identical with a biplanar space [2]. eom.springer.de /b/b016610.htm   (442 words)

5 Bivectors Bivectors can be explained by analogy with vectors as follows: we specify a vector by choosing a line (e.g. A bivector, like a vector, does not have a location, and so the same drawing anywhere on the sheet represents the same bivector. But a bivector differs from a parallelogram in another way as well: a bivector has no shape, and so any patch with the same area and orientation in the same plane is a picture of the same bivector. www.physpharm.fmd.uwo.ca /undergrad/tweedweb/5bivectors.htm   (2276 words)

Bivector   (Site not responding. Last check: 2007-11-03) A bivector is a two-dimensional analog to a one-dimensional represent a one-dimensional directed quantity (often visualized geometrically as a directed line-segment), a bivector is used to represent a two-dimensional directed quantity (often visualized as an oriented plane). orientation of the bivector is established by placing the two vectors tail-to-tail and sweeping from the first vector to the second. 202.41.85.103 /manuals/planetmath/entries/14/Bivector/Bivector.html   (112 words)

Bivector exponentiation - GASP In this we show that bivectors of the form $B=\backslash phi\; P+tn$, where $\backslash phi$ is a scalar, $P$ is a spatial bivector, $t$ is a spatial vector and $n\; =\; e\; +\; \backslash bar\{e\}$, can be exponentiated to give combined rotation and translation rotors via the relation This is used in the above paper to present a new form of pose and position interpolation which is a natural extension of the usual quaternionic SLERP interpolation for rotations into translations and, potentially, non-Euclidean spaces. In Applications of Geometric Algebra (2004) the pose bivector exponential was generalized to include dilation. www-sigproc.eng.cam.ac.uk /ga/index.php?title=Generalized_Bivector_Exponentials   (239 words)

Answer to Pierre’s Puzzle A bivector can be represented as a patch of area, with a direction of circulation marked on it. The direction of circulation of the bivector is such that the edge nearest the wire is directed oppositely to the current in the wire. Even though the metal of which the needle is made has a macroscopically symmetric shape, the magnetic field (when represented as it should be, by a bivector) does not have reflection symmetry. www.av8n.com /physics/pierre-answer.htm   (1093 words)

NTL Like any other NTL container, BiVector is moveable type with pick and optional deep copy transfer semantics. Adds new element at the head of BiVector and picks value of parameter to it. Minimizes memory consumption of BiVector by minimizing capacity. www.volny.cz /cxl/bivector.html   (525 words)

Introduction to Clifford Algebra As a concrete example of addition of bivectors, consider a gyroscopic precession problem, as follows: The green bivector is the initial angular momentum of the system, and the small purple bivector is torque*time. Q behaves exactly as we would expect a bivector to behave, based on the description given in section 1.1: a patch of surface with a direction of circulation around its edge. The sequence would have been: (a) establish a few fundamental notions; (b) set forth the behavior of the basis vectors according to equation 22 and equation 23; (c) express all vectors, bivectors, etc. in terms of their components relative to this basis; and (d) derive the main results in terms of components. www.av8n.com /physics/clifford-intro.htm   (4638 words)

The Geometric Product is a bivector area, the inner and outer products respectively lower and raise the grade of a vector. The result of adding a scalar to a bivector is an object that has both scalar and bivector parts, in exactly the same way that the addition of real and imaginary numbers yields an object with both real and imaginary parts. bivector) as a `multivector', accepting throughout that we are combining objects of different types. www.mrao.cam.ac.uk /~clifford/introduction/intro/node5.html   (443 words)

Amazon.com: "unit bivector": Key Phrase page   (Site not responding. Last check: 2007-11-03) See all pages with references to unit bivector. a unit bivector i proportional to B as the direction of the plane. It is an example of a bivector, the unit bivector. amazon.com /phrase/unit-bivector   (497 words)

The Geometric Algebra of 3D Euclidean Space In particular we generate the unit scalar and we shall prove that also two new kinds of objects different from scalars and vectors are obtained. Objects of the type e1 e2 will be called bivectors and the ones of the type e1 e2 e3 will be called trivectors for obvious reasons. Two pieces with the same area on two parallel planes with the same orientation are represented by the same bivector no matter what the shape of the area looks like or where the planes are located. omega.albany.edu:8008 /mat220dir/ga3d-dir/GA3d.html   (2218 words)

What IS the Hodge Star * Map?   (Site not responding. Last check: 2007-11-03) An SU(2) = Spin(3) bivector 2-vector space acts as a transitive transformation group of the symmetric space Spin(3) / Spin(2) = S2 and S2 x S2 is a 4-dimensional space with quaternionic structure. A Spin(8) bivector 2-vector space acts as a transitive transformation group of the symmetric space Spin(8) / Spin(7) = S7 and S7 x RP1 is an 8-dimensional space with octonionic structure. However, a Spin(8) bivector 2-vector space is too big to act as a transitive transformation group of a symmetric space of the form Spin(8) / G = M where the dimension of M is 4 or less. www.valdostamuseum.org /hamsmith/hstar.html   (962 words)

Maths - Maths - 3D Multivector basis - Martin Baker - Martin Baker Or more correctly a bivector because it has slightly different properties in that the result of cross product with bivector operands can be reversed in certain circumstances. We could use the direction of the bivector to define the plane and the length of the bivector to define the angle, in this way the the bivector alone can fully define a rotation. The vector cross product gives a vector (or more strictly - a bivector) which is perpendicular to both the vectors being multiplied. www.euclideanspace.com /maths/algebra/clifford/d3/geometry/index.htm   (986 words)

Apparatus for operating double vector and encrypting system including the same - Patent 6560336 1 is a block diagram of an apparatus for summing bivectors, in accordance with an embodiment of the present invention. 8 is a block diagram of an apparatus for doubling a bivector, in accordance with an embodiment of the present invention. 10 is a block diagram of an apparatus for multiplying a bivector by an integer, in accordance with an embodiment of the present invention. www.freepatentsonline.com /6560336.html   (8955 words)

cl2.nb A vector with grade 2 is called bivector which will be defined in the context of the product of two vectors. The effect on a bivector causes a reversion, which at the end changes the sign of the bivector. The bivector part gives the are spanned by the two vectors multiplied by the unit bivector. socr.uwindsor.ca /~cabrer7/tclifford/cl2   (882 words)

IngentaConnect Dirac theory in spacetime algebra: I. The generalized bivector Di...   (Site not responding. Last check: 2007-11-03) IngentaConnect Dirac theory in spacetime algebra: I. The generalized bivector Di... Spinor fields mix bivectors and vectors which have different properties in spacetime algebra. Instead the Dirac field is formulated as a generalized bivector field. www.ingentaconnect.com /content/iop/jphysa/2001/00000034/00000010/art00304   (222 words)

Maths - Inverse Vector - Martin Baker In three dimensions the bivector has similar properties to a vector (they are duals) and so, for most purposes, can be treated as a vector. There are some differences, for instance a bivector squares to a negative number, whereas a vector squares to a positive number and there are sign changes when a bivector is multiplied by a vector. This duality only applies to three dimensions, if two 2-dimentional vectors are cross multiplied the bivector is one dimensional, if two 4-dimentional vectors are cross multiplied the bivector is 6-dimentional. www.euclideanspace.com /maths/algebra/vectors/vecAlgebra/index.htm   (2559 words)

Amazon.com: "spatial bivector": Key Phrase page   (Site not responding. Last check: 2007-11-03) As we saw in chapter 1, the basic action of a spatial bivector such as e12 on a vector v = vle1 + v2e2 in the same plane is to rotate it by... By expanding (16) in the spatial bivector basis {ia1, ia2,... Where required, relative (or spatial) vectors in the 'yo-system are written in bold type to record the fact that in... www.amazon.com /phrase/spatial-bivector   (582 words)

NTL On the other hand, it provides some special operations impossible for BiVector and most important, it never invalidates references (that means C++ references and pointers) to elements (it often invalidates iterators, though). Disadvantage of BiArray over BiVector is performance - most operations are significantly slower than BiVector's one (by factor up to 8, depends on speed of malloc/free). As for memory, for small size of elements, BiArray memory consumption is significantly worse than BiVector consumption. www.volny.cz /cxl/biarray.html   (968 words)

CLUCalc: CLUCalc - A Visual Calculator Switch to set visualization of a bivector to a circular plane. Use this variable to switch the drawing mode of Euclidean bivector to circular planes. Use this variable to switch the drawing mode of Euclidean bivector to rectangular planes. www.perwass.de /CLU/CLUCalcDoc/group__grp__MVStype__E3.html   (214 words)

2 Introduction to geometric algebra This bivector is the geometric product or, quite simply, the product; this product is distributive. The exponential of bivectors is useful for defining rotations; a rotation of vector a by angle θ on the σ It is more general to define a rotation by a plane (bivector) then by an axis (vector) because the latter only works in 3D while the former is applicable in any dimension. bda.planetaclix.pt /pirt/4do06_3se2.html   (1297 words)

Angmom web page Parallelogram with sides r and v is the bivector r /\ v. r /\ v defines orbital plane because the bivector lies in it. the bivector is a rectangle and its area is simply clowder.net /hop/angmom/angmom.html   (137 words)

Amazon.com: "unit bivector": Key Phrase page   (Site not responding. Last check: 2007-11-03) a unit bivector i proportional to B as the direction of the plane. It is an example of a bivector, the unit bivector. The unit bivector squares to -1 and generates rotations through 90. www.amazon.com /phrase/unit-bivector   (483 words)

Non-Noether symmetries in Hamiltonian Dynamical Systems Lutzky's theorem is reformulated in terms of bivector fields and alternative derivation of conserved quantities suitable for computations in infinite dimensional Hamiltonian dynamical systems is suggested. In terms of bivector fields these bi-Hamiltonian system is formed by The conservation laws (45) associated with the symmetry reproduce well known set of conservation laws of Toda chain. However Hamiltonian realization of this equation is unknown (for instance Poisson bivector field of dispersive water wave system (252) vanishes during reduction). www.geocities.com /chavchan/21/xml/ham.xml   (6977 words)

The Bivector Avoids Scaling Difficulties.   (Site not responding. Last check: 2007-11-03) : : Only you have been confusing (not conflating) bivectors, vectors, areas, : and apparently a large number of other concepts. The failure of scaling (or metric) invariance is only the start of the difficulties, as can be shown by elementary logic. The scaling failure, and about every other compounding difficulty, disappears when an oriented area is represented by a bivector. kyoto.cool.ne.jp /mburns/vectore1.html   (267 words)

Geometric algebra - CGAFaq   (Site not responding. Last check: 2007-11-03) Every term in the final sum is a bivector, the two y y terms have cancelled out what would have been a scalar portion. In general every 3D bivector is, as in the example, a linear combination of a bivector basis we write as x∧y, y∧z, z∧x. Only in 3D is the cross product available, by dualizing the bivector result of the wedge product. cgafaq.info /index.php?title=Geometric_algebra&redirect=no   (905 words)

bivector - OneLook Dictionary Search We found 9 dictionaries with English definitions that include the word bivector: Tip: Click on the first link on a line below to go directly to a page where "bivector" is defined. Bivector : Eric Weisstein's World of Mathematics [home, info] www.onelook.com /?w=bivector   (142 words)

Bivector This is the definition of the term Bivector Bivector (n.) A term made up of the two parts / + /1 /-1, where / and /1 are vectors. For people who have trouble spelling, this is the defintion of the term Bivector linkspider.serversystems.net /dictionary/lookup/bivector   (79 words)

Projection & rejection from/on bivector   (Site not responding. Last check: 2007-11-03) Projection and rejection of vector from/on a bivector r = rB/B = (r^B)/B = (x^B)/B. It is instructive to compare the projection (x.B)/B and the rejection (x^B)/B with respect to the bivector B to the projection (x.a)/a and the rejection (x^a)/a with respect to a vector a. You can change the vector x by interactively dragging the bright red point with the mouse. sinai.mech.fukui-u.ac.jp /gcj/software/GAcindy/bivectors/Brepro.html   (72 words)

Quaternion Quantum Mechanics It is possible to avoid the imaginary unit i in all equations including the Dirac equation with the help of these bivector units. The imaginary unit is mathematically convenient however, and most people are used to seeing it.  For that reason, what I’ve done is to group i with the appropriate line vector. i a is the unit bivector defining the plane of rotation) home.pcisys.net /~bestwork.1/QRW/QuaternionQuantumMechanics.htm   (959 words)

Non-Noether symmetries and their influence on phase space geometry   (Site not responding. Last check: 2007-11-03) Skew symmetry of the bivector field W provides the skew symmetry of the corresponding Poisson bracket and the condition (1) ensures that for every triple (f, g, h) of smooth functions on the phase space the Jacobi identity We also assume that the dynamical system under consideration is regular – the bivector field W has maximal rank, i. Thus, we have proved that d and đ are differential operators (in fact d is ordinary exterior differential and the expression (22) is its well known representation in terms of Poisson bivector field). www.rmi.acnet.ge /~gch/presto/samp4m.html   (2913 words)

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