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Chasles' Theorem (kinematics)
In kinematics, Chasles' theorem, or Mozzi–Chasles' theorem, says that the most general rigid body displacement can be produced by a screw displacement. A direct Euclidean isometry in three dimensions involves a translation and a rotation. The screw displacement representation of the isometry decomposes the translation into two components, one parallel to the axis of the rotation associated with the isometry and the other component perpendicular to that axis. The Chasles theorem states that the axis of rotation can be selected to provide the second component of the original translation as a result of the rotation. This theorem in three dimensions extends a similar representation of planar isometries as rotation. Once the screw axis is selected, the screw displacement rotates about it and a translation parallel to the axis is included in the screw displacement. Planar isometries with complex numbers Euclidean geometry is expressed in the complex plane by points p = x + y i where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pure Screw
Pure may refer to: Computing * Pure function * PureSystems, a family of computer systems introduced by IBM in 2012 * Pure Software, a company founded in 1991 by Reed Hastings to support the Purify tool * Pure-FTPd, FTP server software * Pure (programming language), functional programming language based on term rewriting * Pure Storage, a company that makes datacenter storage solutions Companies and products * Pure (app), dating app * Pure (company), a British consumer electronics company specialising in digital radios * Pure (restaurant chain), a British fast food chain * Pure Insurance, Privilege Underwriters Reciprocal Exchange * Pure Trading, a Canadian electronic communication network operated by CNQ * Pure Oil, a U.S. chain of gas stations * Propulsion Universelle et Récuperation d'Énergie (PURE), a motorsport engineering company * Pure FM (Portsmouth), a university radio station based in Portsmouth, UK * Pure (Belgian radio station), a former Belgian radio station Liter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Algebra
In mathematics, a geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division (though generally not by all elements) and addition of objects of different dimensions. The geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related exterior algebra. In 1878, William Kingdon Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chose to call them "geometric algebras"). Clifford defined the Clifford algebra and its product as a unification of the Gras ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kinematics
In physics, kinematics studies the geometrical aspects of motion of physical objects independent of forces that set them in motion. Constrained motion such as linked machine parts are also described as kinematics. Kinematics is concerned with systems of specification of objects' positions and velocities and mathematical transformations between such systems. These systems may be rectangular like Cartesian coordinate system, cartesian, Curvilinear coordinates like polar coordinates or other systems. The object trajectories may be specified with respect to other objects which may themselve be in motion relative to a standard reference. Rotating systems may also be used. Numerous practical problems in kinematics involve constraints, such as mechanical linkages, ropes, or rolling disks. Overview Kinematics is a subfield of physics and mathematics, developed in classical mechanics, that describes the motion of points, Physical object, bodies (objects), and systems of bodies (group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Theorems
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2211
In contemporary history, the third millennium is the current millennium in the ''Anno Domini'' or Common Era, under the Gregorian calendar. It began on 1 January 2001 ( MMI) and will end on 31 December 3000 ( MMM), spanning the 21st to 30th centuries. Ongoing futures studies seek to understand what will likely continue and what could plausibly change in this period and beyond. Predictions and forecasts not included on this timeline * Climate change * Extinction * List of dates predicted for apocalyptic events * List of future astronomical events ** List of lunar eclipses in the 21st century ** List of solar eclipses in the 21st century * List of time capsules * Near future centennial (bi, tri, etc.) events. * Near future in fiction * Predictions and claims for the Second Coming * Projections of population growth ** Representative Concentration Pathway ** Shared Socioeconomic Pathways 21st century 2000s * See: 2000 * 2001 * 2002 * 2003 * 2004 * 2005 * 2006 * 2007 * 20 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard M
Richard is a male given name. It originates, via Old French, from Frankish language, Old Frankish and is a Compound (linguistics), compound of the words descending from Proto-Germanic language, Proto-Germanic ''*rīk-'' 'ruler, leader, king' and ''*hardu-'' 'strong, brave, hardy', and it therefore means 'strong in rule'. Nicknames include "Richie", "Dick (nickname), Dick", "Dickon", "Dickie (name), Dickie", "Rich (given name), Rich", "Rick (given name), Rick", "Rico (name), Rico", "Ricky (given name), Ricky", and more. Richard is a common English (the name was introduced into England by the Normans), German and French male name. It's also used in many more languages, particularly Germanic, such as Norwegian, Danish, Swedish, Icelandic, and Dutch, as well as other languages including Irish, Scottish, Welsh and Finnish. Richard is cognate with variants of the name in other European languages, such as the Swedish "Rickard", the Portuguese and Spanish "Ricardo" and the Italian "Ricc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Internet Archive
The Internet Archive is an American 501(c)(3) organization, non-profit organization founded in 1996 by Brewster Kahle that runs a digital library website, archive.org. It provides free access to collections of digitized media including websites, Application software, software applications, music, audiovisual, and print materials. The Archive also advocates a Information wants to be free, free and open Internet. Its mission is committing to provide "universal access to all knowledge". The Internet Archive allows the public to upload and download digital material to its data cluster, but the bulk of its data is collected automatically by its web crawlers, which work to preserve as much of the public web as possible. Its web archiving, web archive, the Wayback Machine, contains hundreds of billions of web captures. The Archive also oversees numerous Internet Archive#Book collections, book digitization projects, collectively one of the world's largest book digitization efforts. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Van Nostrand
David Van Nostrand (December 5, 1811 – June 14, 1886) was a New York City publisher. Biography David Van Nostrand was born in New York City on December 5, 1811. He was educated at Union Hall, Jamaica, New York, and in 1826 entered the publishing house of John P. Haven, who gave him an interest in the firm when he became of age. In 1834 he formed a partnership with William Dwight, but the financial crisis of 1837 led to its dissolution. Van Nostrand then accepted an appointment as clerk of accounts and disbursements under Captain John G. Barnard, at that time in charge of the defensive works of Louisiana and Texas, with headquarters at New Orleans. While so engaged he devoted attention to the study of scientific and military affairs, and on his return to New York City he began the importation of military books for officers of the U.S. Army, afterward receiving orders from private individuals and from academic institutions for foreign books of science. His place of business ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Benjamin Peirce
Benjamin Peirce (; April 4, 1809 – October 6, 1880) was an American mathematician who taught at Harvard University for approximately 50 years. He made contributions to celestial mechanics, statistics, number theory, algebra, and the philosophy of mathematics. Early life He was born in Salem, Massachusetts, the son of first cousins Benjamin Peirce (1778–1831), later librarian of Harvard, and Lydia Ropes Nichols Peirce (1781–1868). After graduating from Harvard University in 1829, he taught mathematics for two years at the Round Hill School in Northampton, and in 1831 was appointed professor of mathematics at Harvard. He added astronomy to his portfolio in 1842, and remained as Harvard professor until his death. In addition, he was instrumental in the development of Harvard's science curriculum, served as the college librarian, and was director of the United States Coast Survey from 1867 to 1874. In 1842, he was elected as a member of the American Philosophical Society ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant Decomposition
The invariant decomposition is a decomposition of the elements of pin groups \text(p,q,r) into orthogonal commuting elements. It is also valid in their subgroups, e.g. orthogonal, pseudo-Euclidean, conformal, and classical groups. Because the elements of Pin groups are the composition of k oriented reflections, the invariant decomposition theorem readsEvery k-reflection can be decomposed into \lceil k/2 \rceil commuting factors. It is named the invariant decomposition because these factors are the invariants of the k-reflection R \in \text(p,q,r). A well known special case is the Chasles' theorem, which states that any rigid body motion in \text(3) can be decomposed into a rotation around, followed or preceded by a translation along, a single line. Both the rotation and the translation leave two lines invariant: the axis of rotation and the orthogonal axis of translation. Since both rotations and translations are bireflections, a more abstract statement of the theorem reads "Eve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane-based Geometric Algebra
Plane-based geometric algebra is an application of Clifford algebra to modelling planes, lines, points, and rigid transformations. Generally this is with the goal of solving applied problems involving these elements and their intersections, Projection (linear algebra), projections, and their angle from one another in 3D space. Originally growing out of research on spin groups, it was developed with applications to robotics in mind. It has since been applied to machine learning, rigid body dynamics, and computer science, especially computer graphics. It is usually combined with a ''duality'' operation into a system known as "Projective Geometric Algebra", see below. Plane-based geometric algebra takes ''planar reflections'' as basic elements, and constructs all other transformations and geometric objects out of them. Formally: it identifies planar reflections with the ''grade-1'' elements of a Clifford Algebra, that is, elements that are written with a single subscript such as "\bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kinematics
In physics, kinematics studies the geometrical aspects of motion of physical objects independent of forces that set them in motion. Constrained motion such as linked machine parts are also described as kinematics. Kinematics is concerned with systems of specification of objects' positions and velocities and mathematical transformations between such systems. These systems may be rectangular like Cartesian coordinate system, cartesian, Curvilinear coordinates like polar coordinates or other systems. The object trajectories may be specified with respect to other objects which may themselve be in motion relative to a standard reference. Rotating systems may also be used. Numerous practical problems in kinematics involve constraints, such as mechanical linkages, ropes, or rolling disks. Overview Kinematics is a subfield of physics and mathematics, developed in classical mechanics, that describes the motion of points, Physical object, bodies (objects), and systems of bodies (group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
