|
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] |
Elements Of Plane-based Geometric Algebra
Element or elements may refer to: Science * Chemical element, a pure substance of one type of atom * Heating element, a device that generates heat by electrical resistance * Orbital elements, parameters required to identify a specific orbit of one body around another * DNA element, a functional region of DNA, including genes and cis-regulatory elements. Mathematics * Element (category theory), one of the constituents in general category theory * Element (mathematics), one of the constituents of set theory in mathematics * Differential element, an infinitesimally small change of a quantity in an integral * Euclid's ''Elements'', a mathematical treatise on geometry and number theory * An entry, or element, of a matrix Philosophy and religion * Classical elements, ancient beliefs about the fundamental types of matter (earth, air, fire, water) * The elements, a religious term referring to the bread and wine of the Eucharist * ''Godai'' (Japanese philosophy), the basis of the un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Screw Theory
Screw theory is the algebraic calculation of pairs of Vector (mathematics and physics), vectors, also known as ''dual vectors'' – such as Angular velocity, angular and linear velocity, or forces and Moment (physics), moments – that arise in the kinematics and Dynamics (mechanics), dynamics of Rigid body, rigid bodies. Screw theory provides a mathematical formulation for the geometry of lines which is central to rigid body dynamics, where lines form the screw axis, screw axes of spatial movement and the Line of action, lines of action of forces. The pair of vectors that form the Plücker coordinates of a line define a unit screw, and general screws are obtained by multiplication by a pair of real numbers and Vector addition, addition of vectors. Important theorems of screw theory include: the ''transfer principle'' proves that geometric calculations for points using vectors have parallel geometric calculations for lines obtained by replacing vectors with screws; Chasles' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternions
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. The algebra of quaternions is often denoted by (for ''Hamilton''), or in blackboard bold by \mathbb H. Quaternions are not a field, because multiplication of quaternions is not, in general, commutative. Quaternions provide a definition of the quotient of two vectors in a three-dimensional space. Quaternions are generally represented in the form a + b\,\mathbf i + c\,\mathbf j +d\,\mathbf k, where the coefficients , , , are real numbers, and , are the ''basis vectors'' or ''basis elements''. 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, robotics, magnetic resonance imaging and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersecting anywhere inside or outside the figure at a ''center of rotation''. A solid figure has an infinite number of possible axes and angles of rotation, including chaotic rotation (between arbitrary orientation (geometry), orientations), in contrast to rotation around a fixed axis, rotation around a axis. The special case of a rotation with an internal axis passing through the body's own center of mass is known as a spin (or ''autorotation''). In that case, the surface intersection of the internal ''spin axis'' can be called a ''pole''; for example, Earth's rotation defines the geographical poles. A rotation around an axis completely external to the moving body is called a revolution (or ''orbit''), e.g. Earth's orbit around the Sun. The en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horizon
The horizon is the apparent curve that separates the surface of a celestial body from its sky when viewed from the perspective of an observer on or near the surface of the relevant body. This curve divides all viewing directions based on whether it intersects the relevant body's surface or not. The ''true horizon'' is a theoretical line, which can only be observed to any degree of accuracy when it lies along a relatively smooth surface such as that of Earth's oceans. At many locations, this line is obscured by terrain, and on Earth it can also be obscured by life forms such as trees and/or human constructs such as buildings. The resulting intersection of such obstructions with the sky is called the ''visible horizon''. On Earth, when looking at a sea from a shore, the part of the sea closest to the horizon is called the offing. Pronounced, "Hor-I-zon". The true horizon surrounds the observer and it is typically assumed to be a circle, drawn on the surface of a perfectly sph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vanishing Point
A vanishing point is a point (geometry), point on the projection plane, image plane of a graphical perspective, perspective rendering where the two-dimensional perspective projections of parallel (geometry), parallel lines in three-dimensional space appear to converge. When the set of parallel lines is perpendicular to a picture plane, the construction is known as one-point perspective, and their vanishing point corresponds to the station point, oculus, or "eye point", from which the image should be viewed for correct perspective geometry.Kirsti Andersen (2007) ''Geometry of an Art'', p. xxx, Springer, Traditional linear drawings use objects with one to three sets of parallels, defining one to three vanishing points. Italian Renaissance humanism, humanist polymath and architect Leon Battista Alberti first introduced the concept in his treatise on perspective in art, ''De pictura'', written in 1435. Straight Track geometry, railroad tracks are a familiar modern example. Vector ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane At Infinity
In projective geometry, a plane at infinity is the hyperplane at infinity of a three dimensional projective space or to any plane contained in the hyperplane at infinity of any projective space of higher dimension. This article will be concerned solely with the three-dimensional case. Definition There are two approaches to defining the ''plane at infinity'' which depend on whether one starts with a projective 3-space or an affine 3-space. If a projective 3-space is given, the ''plane at infinity'' is any distinguished projective plane of the space. This point of view emphasizes the fact that this plane is not geometrically different than any other plane. On the other hand, given an affine 3-space, the ''plane at infinity'' is a projective plane which is added to the affine 3-space in order to give it closure of incidence properties. Meaning that the points of the ''plane at infinity'' are the points where parallel lines of the affine 3-space will meet, and the lines are the l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elements At Infinity Including Stars In The Sky
Element or elements may refer to: Science * Chemical element, a pure substance of one type of atom * Heating element, a device that generates heat by electrical resistance * Orbital elements, parameters required to identify a specific orbit of one body around another * DNA element, a functional region of DNA, including genes and cis-regulatory elements. Mathematics * Element (category theory), one of the constituents in general category theory * Element (mathematics), one of the constituents of set theory in mathematics * Differential element, an infinitesimally small change of a quantity in an integral * Euclid's ''Elements'', a mathematical treatise on geometry and number theory * An entry, or element, of a matrix Philosophy and religion * Classical elements, ancient beliefs about the fundamental types of matter (earth, air, fire, water) * The elements, a religious term referring to the bread and wine of the Eucharist * ''Godai'' (Japanese philosophy), the basis of the un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Function
Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when is the identity function, the equality is true for all values of to which can be applied. Definition Formally, if is a set, the identity function on is defined to be a function with as its domain and codomain, satisfying In other words, the function value in the codomain is always the same as the input element in the domain . The identity function on is clearly an injective function as well as a surjective function (its codomain is also its range), so it is bijective. The identity function on is often denoted by . In set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or ''diagonal'' of . Algebraic propert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis (mathematics)
In mathematics, a set of elements of a vector space is called a basis (: bases) if every element of can be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to . The elements of a basis are called . Equivalently, a set is a basis if its elements are linearly independent and every element of is a linear combination of elements of . In other words, a basis is a linearly independent spanning set. A vector space can have several bases; however all the bases have the same number of elements, called the dimension of the vector space. This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. Basis vectors find applications in the study of crystal structures and frames of reference. Definition A basis of a vector space over a field (such as th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pga Elements
PGA is an acronym or initialism that may stand for: Aviation * IATA code for Page Municipal Airport, Coconino County, Arizona * ICAO designator for Portugália, regional airline based in Lisbon, Portugal * Abbreviation for Prince George Airport, British Columbia, Canada Organizations * Parliamentarians for Global Action, an international parliamentary group that engage in a range of action-oriented initiatives. * Peoples' Global Action, a worldwide co-ordination of radical social movements * Producers Guild of America, an organization representing television producers, film producers and new media producers in the United States Golf Organizations and tours * Professional Golfers' Association (Great Britain and Ireland) * Professional Golfers' Association of America * PGA of Australia * PGA Tour, United States–based organization (independent of the PGA of America) that operates men's professional golf tours, and the name of the elite tour it runs * PGA European Tour, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
