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Examples Of Polar Coordinates
In geometry , a COORDINATE SYSTEM is a system which uses one or more numbers , or COORDINATES, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space . The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics , but may be complex numbers or elements of a more abstract system such as a commutative ring . The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry
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Canonical Coordinates
In mathematics and classical mechanics , CANONICAL COORDINATES are sets of coordinates which can be used to describe a physical system at any given point in time (locating the system within phase space ). Canonical coordinates are used in the Hamiltonian formulation of classical mechanics . A closely related concept also appears in quantum mechanics ; see the Stone–von Neumann theorem and canonical commutation relations for details. As Hamiltonian mechanics is generalized by symplectic geometry and canonical transformations are generalized by contact transformations , so the 19th century definition of canonical coordinates in classical mechanics may be generalized to a more abstract 20th century definition of coordinates on the cotangent bundle of a manifold
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Lagrangian Mechanics
LAGRANGIAN MECHANICS is a reformulation of classical mechanics , introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788. In Lagrangian mechanics, the trajectory of a system of particles is derived by solving the Lagrange equations in one of two forms, either the Lagrange equations of the first kind, which treat constraints explicitly as extra equations, often using Lagrange multipliers ; or the Lagrange equations of the second kind, which incorporate the constraints directly by judicious choice of generalized coordinates . In each case, a mathematical function called the LAGRANGIAN is a function of the generalized coordinates, their time derivatives, and time, and contains the information about the dynamics of the system. No new physics is introduced in Lagrangian mechanics
Lagrangian mechanics
compared to Newtonian mechanics
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Hamiltonian Mechanics
HAMILTONIAN MECHANICS is a theory developed as a reformulation of classical mechanics and predicts the same outcomes as non-Hamiltonian classical mechanics. It uses a different mathematical formalism, providing a more abstract understanding of the theory. Historically, it was an important reformulation of classical mechanics, which later contributed to the formulation of statistical mechanics and quantum mechanics . Hamiltonian mechanics was first formulated by William Rowan Hamilton in 1833, starting from Lagrangian mechanics , a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788
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Barycentric Coordinates (mathematics)
In geometry , the BARYCENTRIC COORDINATE SYSTEM is a coordinate system in which the location of a point of a simplex (a triangle, tetrahedron , etc.) is specified as the center of mass , or barycenter, of usually unequal masses placed at its vertices . Coordinates also extend outside the simplex, where one or more coordinates become negative . The system was introduced (1827) by August Ferdinand Möbius
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Ternary Plot
A TERNARY PLOT, TERNARY GRAPH, TRIANGLE PLOT, SIMPLEX PLOT, GIBBS TRIANGLE or DE FINETTI DIAGRAM is a barycentric plot on three variables which sum to a constant. It graphically depicts the ratios of the three variables as positions in an equilateral triangle . It is used in physical chemistry , petrology , mineralogy , metallurgy , and other physical sciences to show the compositions of systems composed of three species. In population genetics , it is often called a de Finetti diagram . In game theory , it is often called a simplex plot. Approximate colours of Ag–Au–Cu alloys in jewellery making In a ternary plot, the proportions of the three variables a, b, and c must sum to some constant, K. Usually, this constant is represented as 1.0 or 100%
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Generalized Coordinates
In analytical mechanics , specifically the study of the rigid body dynamics of multibody systems , the term GENERALIZED COORDINATES refers to the parameters that describe the configuration of the system relative to some reference configuration. These parameters must uniquely define the configuration of the system relative to the reference configuration. This is done assuming that this can be done with a single chart . The GENERALIZED VELOCITIES are the time derivatives of the generalized coordinates of the system. An example of a generalized coordinate is the angle that locates a point moving on a circle. The adjective "generalized" distinguishes these parameters from the traditional use of the term coordinate to refer to Cartesian coordinates
Cartesian coordinates
: for example, describing the location of the point on the circle using x and y coordinates
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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 , P3. Because they satisfy a quadratic constraint, they establish a one-to-one correspondence between the 4-dimensional space of lines in P3 and points on a quadric in P5 (projective 5-space). A predecessor and special case of Grassmann coordinates (which describe k-dimensional linear subspaces, or flats, in an n-dimensional Euclidean space
Euclidean space
), Plücker coordinates
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
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Orthogonal Coordinates
In mathematics , ORTHOGONAL COORDINATES are defined as a set of d coordinates Q = (q1, q2, ..., qd) in which the coordinate surfaces all meet at right angles (note: superscripts are indices , not exponents). A coordinate surface for a particular coordinate qk is the curve, surface, or hypersurface on which qk is a constant. For example, the three-dimensional Cartesian coordinates
Cartesian coordinates
(x, y, z) is an orthogonal coordinate system, since its coordinate surfaces x = constant, y = constant, and z = constant are planes that meet at right angles to one another, i.e., are perpendicular. Orthogonal
Orthogonal
coordinates are a special but extremely common case of curvilinear coordinates
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Coordinate Surfaces
In geometry , a COORDINATE SYSTEM is a system which uses one or more numbers , or COORDINATES, to uniquely determine the position of a point or other geometric element on a manifold such as Euclidean space . The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics , but may be complex numbers or elements of a more abstract system such as a commutative ring . The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry
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Skew Coordinates
A system of SKEW COORDINATES is a curvilinear coordinate system where the coordinate surfaces are not orthogonal , in contrast to orthogonal coordinates . Skew coordinates
Skew coordinates
tend to be more complicated to work with compared to orthogonal coordinates since the metric tensor will have nonzero off-diagonal components, preventing many simplifications in formulas for tensor algebra and tensor calculus . The nonzero off-diagonal components of the metric tensor are a direct result of the non-orthogonality of the basis vectors of the coordinates, since by definition: g i j = e i e j {displaystyle g_{ij}=mathbf {e} _{i}cdot mathbf {e} _{j}} where g i j {displaystyle g_{ij}} is the metric tensor and e i {displaystyle mathbf {e} _{i}} the (covariant) basis vectors . These coordinate systems can be useful if the geometry of a problem fits well into a skewed system
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Log-polar Coordinates
In mathematics , LOG-POLAR COORDINATES (or LOGARITHMIC POLAR COORDINATES) is a coordinate system in two dimensions, where a point is identified by two numbers, one for the logarithm of the distance to a certain point, and one for an angle . Log-polar coordinates
Log-polar coordinates
are closely connected to polar coordinates , which are usually used to describe domains in the plane with some sort of rotational symmetry . In areas like harmonic and complex analysis , the log-polar coordinates are more canonical than polar coordinates
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Triangle
A TRIANGLE is a polygon with three edges and three vertices . It is one of the basic shapes in geometry . A triangle with vertices A, B, and C is denoted A B C {displaystyle triangle ABC} . In Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space ). This article is about triangles in Euclidean geometry except where otherwise noted
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Trilinear Coordinates
In geometry , the TRILINEAR COORDINATES x:y:z of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates
Trilinear coordinates
are an example of homogeneous coordinates . They are often called simply "trilinears". The ratio x:y is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices A and B respectively; the ratio y:z is the ratio of the perpendicular distances from the point to the sidelines opposite vertices B and C respectively; and likewise for z:x and vertices C and A. In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances (a' , b' , c' ), or equivalently in ratio form, ka' :kb' :kc' for any positive constant k. If a point is on a sideline of the reference triangle, its corresponding trilinear coordinate is 0
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Duality (mathematics)
In mathematics , a DUALITY, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points , so that the dual of A is A itself. For example, Desargues\' theorem is self-dual in this sense under the standard duality in projective geometry . In mathematical contexts, duality has numerous meanings although it is "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics". Many mathematical dualities between objects of two types correspond to pairings , bilinear functions from an object of one type and another object of the second type to some family of scalars
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Line Coordinates
In geometry , LINE COORDINATES are used to specify the position of a line just as point coordinates (or simply coordinates ) are used to specify the position of a point. CONTENTS * 1 Lines in the plane * 2 Tangential equations * 3 Tangential equation of a point * 4 Formulas * 5 Lines in three-dimensional space * 6 With complex numbers * 7 See also * 8 References LINES IN THE PLANEThere are several possible ways to specify the position of a line in the plane. A simple way is by the pair (m, b) where the equation of the line is y = mx + b. Here m is the slope and b is the y-intercept . This system specifies coordinates for all lines that are not vertical. However, it is more common and simpler algebraically to use coordinates (l, m) where the equation of the line is lx + my + 1 = 0. This system specifies coordinates for all lines except those that pass through the origin
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