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Quaternion (other)
In mathematics The quaternions form a number system that extends the complex numbers. * Quaternion rotation * Quaternion group, a non-abelian group of order 8 Symbols * Imperial quaternions (heraldry of the Holy Roman Empire) ** Quaternion Eagle Military uses *A group of four soldiers in the Roman legion *A fireteam Other * Quaternion (gathering), four folded sheets as a unit in bookbinding * Quaternion (poetry) Quaternion is a poetry style in which the theme is divided into four parts. Characteristics Each part of a quaternion explores the complementary natures of the theme or subject. The word ''quaternion'' is derived from the Latin word ''quaterni'', m ..., a style of poetry with four parts See also * {{disambiguation, maths ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 them ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternion 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] |
Quaternion Group
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset \ of the quaternions under multiplication. It is given by the group presentation :\mathrm_8 = \langle \bar,i,j,k \mid \bar^2 = e, \;i^2 = j^2 = k^2 = ijk = \bar \rangle , where ''e'' is the identity element and commutes with the other elements of the group. Another presentation of Q8 is :\mathrm_8 = \langle a,b \mid a^4 = e, a^2 = b^2, ba = a^b\rangle. Compared to dihedral group The quaternion group Q8 has the same order as the dihedral group D4, but a different structure, as shown by their Cayley and cycle graphs: In the diagrams for D4, the group elements are marked with their action on a letter F in the defining representation R2. The same cannot be done for Q8, since it has no faithful representation in R2 or R3. D4 can be realized as a subset of the split-quaternions in the same way that Q8 can be viewed as a sub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imperial Quaternions
An Imperial State or Imperial Estate ( la, Status Imperii; german: Reichsstand, plural: ') was a part of the Holy Roman Empire with representation and the right to vote in the Imperial Diet ('). Rulers of these Estates were able to exercise significant rights and privileges and were " immediate", meaning that the only authority above them was the Holy Roman Emperor. They were thus able to rule their territories with a considerable degree of autonomy. The system of imperial states replaced the more regular division of Germany into stem duchies in the early medieval period. The old Carolingian stem duchies were retained as the major divisions of Germany under the Salian dynasty, but they became increasingly obsolete during the early high medieval period under the Hohenstaufen, and they were finally abolished in 1180 by Frederick Barbarossa in favour of more numerous territorial divisions. From 1489, the imperial Estates represented in the Diet were divided into three chamb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternion Eagle
The Quaternion Eagle (german: Quaternionenadler; it, aquila quaternione), also known as the Imperial Quaternion Eagle (german: Quaternionen-Reichsadler) or simply Imperial Eagle (german: Reichsadler), was an informal coat of arms of the Holy Roman Empire. Introduced around 1510 by Hans Burgkmair, the Quaternion Eagle mixed two pre-existing concepts: the Imperial Quaternions and the Imperial Eagle (double-headed eagle). History Background The so-called imperial quaternions (german: Quaternionen der Reichsverfassung, , quaternions of the imperial constitution; from la, quaterniō, , group of four soldiers) were a conventional representation of the Imperial States of the Holy Roman Empire which first became current in the 15th century and was extremely popular during the 16th century. Apart from the highest tiers of the emperor, kings, prince-bishops and the prince electors, the estates are represented in groups of four. The number of quaternions was usually ten, in d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roman Legion
The Roman legion ( la, legiō, ) was the largest military unit of the Roman army, composed of 5,200 infantry and 300 equites (cavalry) in the period of the Roman Republic (509 BC–27 BC) and of 5,600 infantry and 200 auxilia in the period of the Roman Empire (27 BC – AD 476). Size The size of a typical legion varied throughout the history of ancient Rome, with complements ranging from 4,200 legionaries and 300 equites (drawn from the wealthier classes – in early Rome all troops provided their own equipment) in the Republican period of Rome (the infantry were split into 10 cohorts each of four maniples of 120 legionaries), to 4,800 legionaries (in 10 cohorts of 6 centuries of 80 legionaries) during Caesar's age, to 5,280 men plus 120 auxiliaries in the Imperial period (split into 10 cohorts, nine of 480 men each, with the first cohort being double-strength at 960 men). It should be noted the above numbers are typical field strengths while "paper strength" was sli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fireteam
A fireteam or fire team is a small military sub-subunit of infantry designed to optimize "bounding overwatch" and "fire and movement" tactical doctrine in combat. Depending on mission requirements, a typical fireteam consists of four or fewer members: an automatic rifleman, a grenadier, a rifleman, and a designated team leader. The role of each fireteam leader is to ensure that the fireteam operates as a cohesive unit. Two or three fireteams are organized into a section or squad in co-ordinated operations, which is led by a squad leader. Historically, nations with effective fireteam organization have had a significantly better performance from their infantry units in combat than those limited to operations by traditionally larger units. US Army doctrine recognizes the fire team, or crew, as the smallest military organization while NATO doctrine refers to this level of organization simply as team. Fireteams are the most basic organization upon which modern infantry units are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternion (gathering)
In bookbinding, a section, gathering, or signature is a group of sheets folded in half, to be worked into the binding as a unit. The section is the basic building block of codex bindings. In Western bookbinding, sections are sewn through their folds, with the sewing thread securing each section to the one bound before it. The gatherings can be seen by looking at the top or bottom sides of the book, though cheaper modern books are perfect bound with no gatherings, each sheet glued directly to the binding. The gatherings are sewn together at the spine, done in such a way that two or more stretches of thread are visible along each gathering's innermost fold. In medieval manuscripts, a gathering, or quire, was most often formed of four folded sheets of vellum or parchment, i.e. 8 leaves, 16 sides. The term ''quaternion'' (or sometimes ''quaternum'') designates such a unit. A gathering made of a single folded sheet (i.e. two leaves, four sides) is a ''bifolium'' (plural ''bifolia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
