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Conformal Geometric Algebra
Conformal geometric algebra (CGA) is the geometric algebra constructed over the resultant space of a map from points in an -dimensional base space to null vectors in . This allows operations on the base space, including reflections, rotations and translations to be represented using versors of the geometric algebra; and it is found that points, lines, planes, circles and spheres gain particularly natural and computationally amenable representations. The effect of the mapping is that generalized (i.e. including zero curvature) -spheres in the base space map onto -blades, and so that the effect of a translation (or ''any'' conformal mapping) of the base space corresponds to a rotation in the higher-dimensional space. In the algebra of this space, based on the geometric product of vectors, such transformations correspond to the algebra's characteristic sandwich operations, similar to the use of quaternions for spatial rotation in 3D, which combine very efficiently. A consequen ... [...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] |
Point At Infinity
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Adjoining these points produces a projective plane, in which no point can be distinguished, if we "forget" which points were added. This holds for a geometry over any field, and more generally over any division ring. In the real case, a point at infinity completes a line into a topologically closed curve. In higher dimensions, all the points at infinity form a projective subspace of one dimension less than that of the whole projective space to which they belong. A point at infinity can also be added to the complex line (which may be thought of as the complex plane), thereby turning it into a closed surface known as the complex projective line, CP1, also called the Riemann sphere (when complex numbers are mapped to each point). In the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Geometry
In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two dimensions, conformal geometry may refer either to the study of conformal transformations of what are called "flat spaces" (such as Euclidean spaces or spheres), or to the study of conformal manifolds which are Riemannian or pseudo-Riemannian manifolds with a class of metrics that are defined up to scale. Study of the flat structures is sometimes termed Möbius geometry, and is a type of Klein geometry. Conformal manifolds A conformal manifold is a Riemannian manifold (or pseudo-Riemannian manifold) equipped with an equivalence class of metric tensors, in which two metrics ''g'' and ''h'' are equivalent if and only if :h = \lambda^2 g , where ''λ'' is a real-valued smooth function defined on the manifold and is called the conformal fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Applied Clifford Algebras
''Advances in Applied Clifford Algebras'' is a peer-reviewed scientific journal that publishes original research papers and also notes, expository and survey articles, book reviews, reproduces abstracts and also reports on conferences and workshops in the area of Clifford algebras and their applications to other branches of mathematics and physics, and in certain cognate areas. There is a vibrant and interdisciplinary community around Clifford and Geometric Algebras with a wide range of applications. The main conferences in this subject includThe International Conference on Clifford Algebras and Their Applications in Mathematical Physics (ICCA)anApplications of Geometric Algebra in Computer Science and Engineering (AGACSE)series. The journal was established in 1991 by Jaime Keller who was its editor-in-chief until his death in 2011. The second editor-in-chief of the journal was Waldyr Alves Rodrigues Jr. (Universidade Estadual de Campinas), and the current editor-in-chief is Uwe K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dilation (metric Space)
In mathematics, a dilation is a function f from a metric space M into itself that satisfies the identity :d(f(x),f(y))=rd(x,y) for all points x, y \in M, where d(x, y) is the distance from x to y and r is some positive real number. In Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ..., such a dilation is a similarity of the space. Dilations change the size but not the shape of an object or figure. Every dilation of a Euclidean space that is not a congruence has a unique fixed point that is called the ''center of dilation''. Some congruences have fixed points and others do not.. See also * Homothety * Dilation (operator theory) References {{DEFAULTSORT:Dilation (Metric Space) Metric geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville's Theorem (conformal Mappings)
In mathematics, Liouville's theorem, proved by Joseph Liouville in 1850, is a rigidity (mathematics), rigidity theorem about conformal mappings in Euclidean space. It states that every smooth function, smooth conformal mapping on a domain of R, where ''n'' > 2, can be expressed as a composition of translation (geometry), translations, similarity (geometry), similarities, orthogonal matrix, orthogonal transformations and inversive geometry#In higher dimensions, inversions: they are Möbius transformation#Higher dimensions, Möbius transformations (in ''n'' dimensions).Philip Hartman (1947Systems of Total Differential Equations and Liouville's theorem on Conformal MappingAmerican Journal of Mathematics 69(2);329–332. This theorem severely limits the variety of possible conformal mappings in R and higher-dimensional spaces. By contrast, conformal mappings in R can be much more complicated – for example, all simply connected planar domains are conformally equivalent, by the Riemann ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Transformation
In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points. The rigid transformations include rotations, translations, reflections, or any sequence of these. Reflections are sometimes excluded from the definition of a rigid transformation by requiring that the transformation also preserve the handedness of objects in the Euclidean space. (A reflection would not preserve handedness; for instance, it would transform a left hand into a right hand.) To avoid ambiguity, a transformation that preserves handedness is known as a rigid motion, a Euclidean motion, or a proper rigid transformation. In dimension two, a rigid motion is either a translation or a rotation. In dimension three, every rigid motion can be decomposed as the composition of a rotation and a translation, and is thus sometimes called a rototranslation. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inversion Transformation
In mathematical physics, inversion transformations are a natural extension of Poincaré transformations to include all conformal map, conformal, bijection, one-to-one transformations on coordinate space-time. They are less studied in physics because, unlike the rotations and translations of Poincaré symmetry, an object cannot be physically transformed by the inversion symmetry. Some physical theories are invariant under this symmetry, in these cases it is what is known as a 'hidden symmetry'. Other hidden symmetries of physics include gauge symmetry and general covariance. Early use In 1831 the mathematician Ludwig Immanuel Magnus began to publish on transformations of the plane generated by inversion in a circle of radius ''R''. His work initiated a large body of publications, now called inversive geometry. The most prominently named mathematician became August Ferdinand Möbius once he reduced the planar transformations to complex number arithmetic. In the company of physicists ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Orthogonal Distance
In geometry, the perpendicular distance between two objects is the distance from one to the other, measured along a line that is perpendicular to one or both. The distance from a point to a line is the distance to the nearest point on that line. That is the point at which a segment from it to the given point is perpendicular to the line. Likewise, the distance from a point to a curve is measured by a line segment that is perpendicular to a tangent line to the curve at the nearest point on the curve. The distance from a point to a plane is measured as the length from the point along a segment that is perpendicular to the plane, meaning that it is perpendicular to all lines in the plane that pass through the nearest point in the plane to the given point. Other instances include: *'' Point on plane closest to origin'', for the perpendicular distance from the origin to a plane in three-dimensional space *'' Nearest distance between skew lines'', for the perpendicular distance b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorentz Transformation
In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity. The transformations are named after the Dutch physicist Hendrik Lorentz. The most common form of the transformation, parametrized by the real constant v, representing a velocity confined to the -direction, is expressed as \begin t' &= \gamma \left( t - \frac \right) \\ x' &= \gamma \left( x - v t \right)\\ y' &= y \\ z' &= z \end where and are the coordinates of an event in two frames with the spatial origins coinciding at , where the primed frame is seen from the unprimed frame as moving with speed along the -axis, where is the speed of light, and \gamma = \frac is the Lorentz factor. When speed is much smal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat (geometry)
In geometry, a flat is an affine subspace, i.e. a subset of an affine space that is itself an affine space. Particularly, in the case the parent space is Euclidean, a flat is a Euclidean subspace which inherits the notion of distance from its parent space. In an -dimensional space, there are -flats of every dimension from 0 to ; flats one dimension lower than the parent space, -flats, are called '' hyperplanes''. The flats in a plane (two-dimensional space) are points, lines, and the plane itself; the flats in three-dimensional space are points, lines, planes, and the space itself. The definition of flat excludes non-straight curves and non-planar surfaces, which are subspaces having different notions of distance: arc length and geodesic length, respectively. Flats occur in linear algebra, as geometric realizations of solution sets of systems of linear equations. A flat is a manifold and an algebraic variety, and is sometimes called a ''linear manifold'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
