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Multivector Field
In multilinear algebra, a multivector, sometimes called Clifford number or multor, is an element of the exterior algebra of a vector space . This algebra is graded algebra, graded, associative algebra, associative and alternating algebra, alternating, and consists of linear combinations of simple -vectors (also known as decomposable -vectors or Blade (geometry), -blades) of the form : v_1\wedge\cdots\wedge v_k, where v_1, \ldots, v_k are in . A -vector is such a linear combination that is ''homogeneous'' of degree (all terms are -blades for the same ). Depending on the authors, a "multivector" may be either a -vector or any element of the exterior algebra (any linear combination of -blades with potentially differing values of ). In differential geometry, a -vector is usually a vector in the exterior algebra of the tangent space, tangent vector space of a smooth manifold; that is, it is an antisymmetric tensor obtained by taking linear combinations of the exterior product of ta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Multilinear Algebra
Multilinear algebra is the study of Function (mathematics), functions with multiple vector space, vector-valued Argument of a function, arguments, with the functions being Linear map, linear maps with respect to each argument. It involves concepts such as Matrix (mathematics), matrices, tensors, multivectors, System of linear equations, systems of linear equations, Higher-dimensional space, higher-dimensional spaces, Determinant, determinants, inner product, inner and outer product, outer products, and Dual space, dual spaces. It is a mathematical tool used in engineering, machine learning, physics, and mathematics. Origin While many theoretical concepts and applications involve Vector space, single vectors, mathematicians such as Hermann Grassmann considered structures involving pairs, triplets, and multivectors that generalize Vector (mathematics and physics), vectors. With multiple combinational possibilities, the space of multivectors expands to 2''n'' dimensions, where ''n'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Scalar (mathematics)
A scalar is an element of a field which is used to define a ''vector space''. In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector. Generally speaking, a vector space may be defined by using any field instead of real numbers (such as complex numbers). Then scalars of that vector space will be elements of the associated field (such as complex numbers). A scalar product operation – not to be confused with scalar multiplication – may be defined on a vector space, allowing two vectors to be multiplied in the defined way to produce a scalar. A vector space equipped with a scalar product is called an inner product space. A quantity described by multiple scalars, such as having both direction and magnitude, is called a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Eric Lengyel
Eric Lengyel is a computer scientist specializing in game engine development, computer graphics, and geometric algebra. He holds a Ph.D. in computer science from the University of California, Davis and a master's degree in mathematics from Virginia Tech, where he also competed in cross-country and track and field. Lengyel is an expert in font rendering technology for 3D applications and is the inventor of the Slug font rendering algorithm, which allows glyphs to be rendered directly from outline data on the GPU with full resolution independence. Lengyel is also the inventor of the Transvoxel algorithm, which is used to seamlessly join multiresolution voxel data at boundaries between different levels of detail that have been triangulated with the Marching cubes algorithm. Among his many written contributions to the field of game development, Lengyel is the author of the four-volume book series ''Foundations of Game Engine Development''. The first volume, covering the mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Plücker Coordinates
In geometry, Plücker coordinates, introduced by Julius Plücker in the 19th century, are a way to assign six homogeneous coordinates to each line in projective 3-space, . Because they satisfy a quadratic constraint, they establish a one-to-one correspondence between the 4-dimensional space of lines in and points on a quadric in (projective 5-space). A predecessor and special case of Grassmann coordinates (which describe -dimensional linear subspaces, or ''flats'', in an -dimensional Euclidean space), Plücker coordinates arise naturally in geometric algebra. They have proved useful for computer graphics, and also can be extended to coordinates for the screws and wrenches in the theory of kinematics used for robot control. Geometric intuition A line in 3-dimensional Euclidean space is determined by two distinct points that it contains, or by two distinct planes that contain it (a plane-plane intersection). Consider the first case, with points x=(x_1,x_2,x_3) and y=( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Duality (projective Geometry)
In projective geometry, duality or plane duality is a formalization of the striking symmetry of the roles played by Point (geometry), points and Line (geometry), lines in the definitions and theorems of projective planes. There are two approaches to the subject of duality, one through language () and the other a more functional approach through special Map (mathematics), mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a ''duality''. Such a map can be constructed in many ways. The concept of plane duality readily extends to ''space duality'' and beyond that to duality in any finite-dimensional projective geometry. Principle of duality A projective plane may be defined axiomatically as an incidence structure, in terms of a set of ''points'', a set of ''lines'', and an incidence relation that de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
