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Quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two '' directed lines'' in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative. Quaternions are generally represented in the form :a + b\ \mathbf i + c\ \mathbf j +d\ \mathbf k where , and are real numbers; and , and are the ''basic quaternions''. Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternions And Spatial Rotation
Unit quaternions, known as ''versors'', provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation about an arbitrary axis. Rotation and orientation quaternions have applications in computer graphics, Presented at SIGGRAPH '85. computer vision, robotics, navigation, molecular dynamics, flight dynamics, orbital mechanics of satellites, and crystallographic texture analysis. When used to represent rotation, unit quaternions are also called rotation quaternions as they represent the 3D rotation group. When used to represent an orientation (rotation relative to a reference coordinate system), they are called orientation quaternions or attitude quaternions. A spatial rotation around a fixed point of \theta radians about a unit axis (X,Y,Z) that denotes the ''Euler axis'' is given by the quaternion (C, X \, S, Y \, S, Z \, S), where C = \cos(\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Angles
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478PDF/ref> They can also represent the orientation of a mobile frame of reference in physics or the orientation of a general basis in 3-dimensional linear algebra. Alternative forms were later introduced by Peter Guthrie Tait and George H. Bryan intended for use in aeronautics and engineering. Chained rotations equivalence Euler angles can be defined by elemental geometry or by composition of rotations. The geometrical definition demonstrates that three composed ''elemental rotations'' (rotations about the axes of a coordinate system) are always sufficient to reach any target frame. The three elemental rotations may be extrinsic (rotations about the axes ''xyz'' of the original coordinate system, which is assumed to remain motionless), or intri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Rowan Hamilton
Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Ireland, living at Dunsink Observatory. Hamilton's scientific career included the study of geometrical optics, ideas from Fourier analysis, and his work on quaternions which made him one of the founders of modern linear algebra. He made major contributions in optics, classical mechanics and abstract algebra. His work was fundamental to modern theoretical physics, particularly his reformulation of Newtonian mechanics, now called Hamiltonian mechanics. It is now central both to electromagnetism and to quantum mechanics. Early life Hamilton was the fourth of nine children born to Sarah Hutton (1780–1817) and Archibald Hamilton (1778–1819),Bruno (2003) who ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation Matrix
In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end rotates points in the plane counterclockwise through an angle with respect to the positive axis about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates , it should be written as a column vector, and multiplied by the matrix : : R\mathbf = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end \begin x \\ y \end = \begin x\cos\theta-y\sin\theta \\ x\sin\theta+y\cos\theta \end. If and are the endpoint coordinates of a vector, where is cosine and is sine, then the above equations become the trigonometric summation angle formulae. Indeed, a rotation matrix can be seen as the trigonometric summation angle formulae in matrix form. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford Algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford. The most familiar Clifford algebras, the orthogonal Clifford algebras, are also referred to as (''pseudo-'')''Riemannian Clifford algebras'', as distinct from ''symplectic Clifford algebras''.see for ex. Introduction and basic properties A Clifford algebra is a unital associative algebra that contains and is generated by a vector space over a field , where is equipped with a qua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector (geometry)
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a '' directed line segment'', or graphically as an arrow connecting an ''initial point'' ''A'' with a ''terminal point'' ''B'', and denoted by \overrightarrow . A vector is what is needed to "carry" the point ''A'' to the point ''B''; the Latin word ''vector'' means "carrier". It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from ''A'' to ''B''. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Theorem (real Division Algebras)
In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers. According to the theorem, every such algebra is isomorphic to one of the following: * (the real numbers) * (the complex numbers) * (the quaternions). These algebras have real dimension , and , respectively. Of these three algebras, and are commutative, but is not. Proof The main ingredients for the following proof are the Cayley–Hamilton theorem and the fundamental theorem of algebra. Introducing some notation * Let be the division algebra in question. * Let be the dimension of . * We identify the real multiples of with . * When we write for an element of , we tacitly assume that is contained in . * We can consider as a finite-dimensional - vector space. Any element of defines an endomorphism of by left-multiplication, we identify with that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composition Algebra
In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies :N(xy) = N(x)N(y) for all and in . A composition algebra includes an involution called a conjugation: x \mapsto x^*. The quadratic form N(x) = x x^* is called the norm of the algebra. A composition algebra (''A'', ∗, ''N'') is either a division algebra or a split algebra, depending on the existence of a non-zero ''v'' in ''A'' such that ''N''(''v'') = 0, called a null vector. When ''x'' is ''not'' a null vector, the multiplicative inverse of ''x'' is When there is a non-zero null vector, ''N'' is an isotropic quadratic form, and "the algebra splits". Structure theorem Every unital composition algebra over a field can be obtained by repeated application of the Cayley–Dickson construction starting from (if the characteristic of is different from ) or a 2-dimensional composition subalgebra (if ). The po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noncommutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says something like or , the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, ); such operations are ''not'' commutative, and so are referred to as ''noncommutative operations''. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized. A similar property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classification Of Clifford Algebras
In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the structures of finite-dimensional real and complex Clifford algebras for a nondegenerate quadratic form have been completely classified. In each case, the Clifford algebra is algebra isomorphic to a full matrix ring over R, C, or H (the quaternions), or to a direct sum of two copies of such an algebra, though not in a canonical way. Below it is shown that distinct Clifford algebras may be algebra-isomorphic, as is the case of Cl2,0(R) and Cl1,1(R), which are both isomorphic to the ring of two-by-two matrices over the real numbers. Notation and conventions The Clifford product is the manifest ring product for the Clifford algebra, and all algebra homomorphisms in this article are with respect to this ring product. Other products defined within Clifford algebras, such as the exterior product, are not used here. This article uses the (+) sign convention for Clifford multi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blackboard Bold
Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical or near-vertical lines) are doubled. The symbols usually denote number sets. One way of producing blackboard bold is to double-strike a character with a small offset on a typewriter. Thus, they are also referred to as double struck. In typography, such a font with characters that are not solid is called an "inline", "shaded", or "tooled" font. History Origin In some texts, these symbols are simply shown in bold type. Blackboard bold in fact originated from the attempt to write bold letters on blackboards in a way that clearly differentiated them from non-bold letters (by using the edge rather than the point of a chalk). It then made its way back into print form as a separate style from ordinary bold, possibly starting with the original 1965 edition of Gunning and Rossi's textbook on complex analysis. Use in textbooks In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Domain (ring Theory)
In algebra, a domain is a nonzero ring in which implies or .Lam (2001), p. 3 (Sometimes such a ring is said to "have the zero-product property".) Equivalently, a domain is a ring in which 0 is the only left zero divisor (or equivalently, the only right zero divisor). A commutative domain is called an integral domain. Mathematical literature contains multiple variants of the definition of "domain".Some authors also consider the zero ring to be a domain: see Polcino M. & Sehgal (2002), p. 65. Some authors apply the term "domain" also to rngs with the zero-product property; such authors consider ''n''Z to be a domain for each positive integer ''n'': see Lanski (2005), p. 343. But integral domains are always required to be nonzero and to have a 1. Examples and non-examples * The ring Z/6Z is not a domain, because the images of 2 and 3 in this ring are nonzero elements with product 0. More generally, for a positive integer ''n'', the ring Z/''n''Z is a domain if and only ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
