The term triangular coordinates may refer to any of at least three related systems of coordinates in the Euclidean plane a special case of barycentric coordinatesmathematics barycentric coordinates for a triangle, in which case it is known as a ternary plot or areal coordinates , among other names, trilinear coordinates , in which the coordinates of a point in a triangle are its relative distances from the three sides, synergetics coordinates Category Triangle geometry Category Coordinate systems ... more details
Image BiangularCoordinateSystem.JPG right frame Biangular coordinates In mathematics , biangular coordinates are a coordinate system for the plane mathematics plane where math C 1 , math and math C 2 , math are two fixed points, and the position of a point P not on the line math overline C 1C 2 , math is determined by the angles math angle PC 1C 2 , math and math angle PC 2C 1 , math . References http links.jstor.org sici?sici 0002 9890 28191002 2917 3A2 3C34 3ABC 3E2.0.CO 3B2 Q&size LARGE G. B. M. Zerr Biangular Coordinates Naylor, A New Kind of Geometry the Biangular Coordinate System http books.google.com books?id XKILFJgS JEC&pg PA38&dq biangular&lr &as brr 0&hl en&cd 6 v onepage&q biangular&f false J. C. L. Fish.Coordinates Of Elementary Surveying , p.38, READ BOOKS, 2007 George Shoobridge Carr. Formulas and theorems in pure mathematics. p.742. Chelsea Pub. Co., 1970 Howard W. Baeumler. Biangular coordinates. University of Buffalo, 1950 Category Coordinate systems geometry stub cs Biangul rn sou adnice ru zh ... more details
In the mathematics mathematical theory of Riemannian geometry , Fermi coordinates are local coordinates that are adapted to a geodesic . ref Manasse and Misner http link.aip.org link JMAPAQ v4 i6 p735 s1 , Fermi Normal Coordinates and Some Basic Concepts in Differential Geometry . Journal of Mathematical Physics 4 6, 1963. ref More formally, suppose M is an n dimensional Riemannian manifold , math gamma math is a geodesic on math M math , and math p math is a point on math gamma math . Then there exists local coordinates math t,x 2, ldots, x n math around math p math such that For small t , math t,0, ldots, 0 math represents the geodesic near math p math , On math gamma math , the metric tensor is the Euclidean metric, On math gamma math , all Christoffel symbol s vanish. Such coordinates are called Fermi coordinates and are named after the Italian physicist Enrico Fermi . The above properties are only valid on the geodesic. For example, if all Christoffel symbols vanish near math p math , then the manifold is Curvature Curvature of space flat near math p math . Reflist See also Geodesic normal coordinates Christoffel symbols Isothermal coordinates DEFAULTSORT Fermi Coordinates Category Riemannian geometry Category Coordinate systems in differential geometry ... more details
. See also Barycentric coordinatesmathematics Trilinear coordinates Quadray coordinates Notes Reflist ...Expert subject date May 2009 Synergetics coordinates is Clifford Nelson s attempt to describe, from another mathematical point of view, Buckminster Fuller s 60 degree coordinate system for understanding nature. Synergetics Fuller Synergetics is the word Fuller used to label his approach to mathematics . ref Clifford Nelson, http library.wolfram.com infocenter MathSource 600 Buckminster Fuller Notebooks ref Geometric definition A Theory of forms system of synergetics coordinates uses only one type of simplex triangle , tetrahedron , pentachoron , ..., n simplex as space units , and in fact uses a regular simplex, rather like Cartesian coordinates use hypercubes Square geometry square , cube , tesseract , ..., n cube. Image Synergetics coordinate plane.jpg right thumb Synergetics coordinates in two dimensions The n Synergetics coordinates axes are perpendicular to the n defining geometric objects that define a regular simplex 2 end points for line segments, 3 lines for triangles, 4 planes for tetrahedrons etc.. The angles between the directions of the coordinate axes are Arc Cosine 1 n 1 . The coordinates can be positive or negative or zero and so can their sum. The sum of the n coordinates is the edge length of the regular simplex defined by moving the n geometric objects in increments of the height of the n 1 dimensional regular simplex that has an edge length of one. If the sum of the n coordinates is negative the triangle n 3 or tetrahedron n 4 is upside down and inside out. Algebraic examples Regular triangular coordinates are in a Lattice graph grid of equilateral triangles ... tetrahedral coordinates are in a Euclidean space Euclidean 3 space grid of equilateral tetrahedra ... Quadray Coordinates Quadray Coordinates on WikiEducator MathWorld title Synergetics Coordinates ... index.htm Synergetics Coordinates DEFAULTSORT Synergetics Coordinates Category Coordinate systems ... more details
Orphan date September 2011 In mathematics , 6 sphere coordinates are the coordinate system created by Inversive geometry inverting the Cartesian coordinate system Cartesian coordinates across the unit sphere . The three coordinates are math u frac x x 2 y 2 z 2 , quad v frac y x 2 y 2 z 2 , quad w frac z x 2 y 2 z 2 . math Since inversion is its own inverse function inverse , the equations for x , y , and z in terms of u , v , and w are similar math x frac u u 2 v 2 w 2 , quad y frac v u 2 v 2 w 2 , quad z frac w u 2 v 2 w 2 . math References Moon, P. and Spencer, D. E. 6 sphere Coordinates. Fig. 4.07 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York Springer Verlag, pp. 122 123, 1988. MathWorld title 6 sphere coordinates urlname 6 SphereCoordinates http demonstrations.wolfram.com SixSphereCoordinates Six Sphere Coordinates by Michael Schreiber, the Wolfram Demonstrations Project . Category Coordinate systems Category Article Feedback 5 geometry stub ... more details
In mathematics , and more particularly in the field of algebraic geometry , Chow coordinates are a generalization of Pl cker coordinates , applying to m dimensional algebraic varieties of degree d in P sup n sup , that is, n dimensional projective space . They are named for W. L. Chow . A Chow variety is a variety whose points correspond to all cycles of a given projective space of given dimension and degree. To define the Chow coordinates, take the intersection of an algebraic variety Z of degree d and dimension m by linear subspaces U of codimension m . When U is in general position , the intersection will be a finite set of d distinct points. Then the coordinates of the d points of intersection are algebraic functions of the Pl cker coordinates of U, and by taking a symmetric function of the algebraic functions, a homogeneous polynomial known as the Chow form or Cayley form of Z is obtained. The Chow coordinates are then the coefficients of the Chow form. Chow coordinates can generate the smallest field of definition of a divisor. The Chow coordinates define a point in the projective space corresponding to all forms, and the closure of the possible Chow coordinates is called the Chow variety. See also Hilbert scheme , a sort of refinement of the Chow varieties. References citation last1 Chow first1 W. L. last2 van der Waerden. first2 B. L. title Zur algebraische Geometrie IX. journal Math. Ann. volume 113 pages 692 704 year 1937 doi 10.1007 BF01571660 W. V. D. Hodge Hodge and Pedoe , vol. II springer id C c022180 first Val.S. last Kulikov title Chow variety Category Algebraic geometry ... more details
In mathematics and classical mechanics , canonical coordinates are particular sets of coordinates on the phase space , or equivalently, on the cotangent manifold of a manifold . Canonical coordinates arise naturally in physics in the study of Hamiltonian mechanics . As Hamiltonian mechanics is generalized by symplectic geometry and canonical transformation s are generalized by contact transformation s, so the 19th century definition of canonical coordinates in classical mechanics may be generalized ... coordinates as they appear in classical mechanics . A closely related concept also appears .... Definition, in classical mechanics In classical mechanics , canonical coordinates are coordinates ... Hamiltonian formalism. The canonical coordinates satisfy the fundamental Poisson bracket relations math q i, q j 0 qquad p i, p j 0 qquad q i, p j delta ij math Canonical coordinates can be obtained from the generalized coordinates of the Lagrangian mechanics Lagrangian formalism by a Legendre transformation , or from another set of canonical coordinates by a canonical transformation . Definition, on cotangent bundles Canonical coordinates are defined as a special set of coordinates on the cotangent ... with the x s or q s denoting the coordinates on the underlying manifold and the p s denoting the conjugate ... definition of canonical coordinates is any set of coordinates on the cotangent bundle that allow the canonical .... A change of coordinates that preserves this form is a canonical transformation these are a special case of a symplectomorphism , which are essentially a change of coordinates on a symplectic ... the momentum function corresponding to X . In atlas topology local coordinates , the vector field ... math T Q math these coordinates are called the canonical coordinates . Generalized coordinates In Lagrangian mechanics , a different set of coordinates are used, called the generalized coordinates ... on the cotangent bundle, then the generalized coordinates are related to the canonical coordinates ... more details
In differential geometry , normal coordinates at a point p in a differentiable manifold equipped with a torsion tensor symmetric affine connection are a local coordinate system in a neighborhood mathematics ... simplifying local calculations. In normal coordinates associated to the Levi Civita connection of a Riemannian ... geometry states that normal coordinates at a point always exist on a manifold with a symmetric affine connection. In such coordinates the covariant derivative reduces to a partial derivative ... the equivalence principle uses normal coordinates via inertial frame s. Normal coordinates always ... to define normal coordinates for Finsler manifold s harv Busemann 1955 . Geodesic normal coordinates Geodesic normal coordinates are local coordinates on a manifold with an affine connection ... coordinates exist on a normal neighborhood of a point p in M . A normal neighborhood U is a subset ... orthonormal bases exist in the domain of E . Properties The properties of normal coordinates often ... and x sup i sup are normal coordinates on U . Let V be some vector from T sub p sub M with components V sup i sup in local coordinates, and math gamma V math be the geodesic with starting point p and velocity vector V , then math gamma V math is represented in normal coordinates by math gamma V t tV 1, ... , tV n math as long as it is in U . The coordinates of p are 0, ... , 0 In Riemannian normal coordinates at p the components of the Metric tensor Riemannian metric g simplify to math delta ij ... of math g ij math . Polar coordinates On a Riemannian manifold, a normal coordinate system at p facilitates the introduction of a system of spherical coordinates , known as polar coordinates . These are the coordinates ... of r ,&phi with the inverse of the exponential map at p is a polar coordinate system. Polar coordinates ... smooth function &fnof . As a result, the metric in polar coordinates assumes a block diagonal ... Citation last1 Busemann first1 Herbert title On normal coordinates in Finsler spaces id MathSciNet ... more details
Fuller Synergetics Barycentric coordinatesmathematics Trilinear coordinates Synergetics coordinates ...Quadray coordinates , also known as tetray coordinates or Chakovian coordinates , were developed by David Chako, Tom Ace, Kirby Urner, Darrel Jarmusch et al., as another take on simplicial coordinates, a coordinate system using the simplex or tetrahedron as its basis polyhedron. ref Urner, Kirby. Teaching Object Oriented Programming with Visual FoxPro. FoxPro Advisor Advisor Media, March, 1999 , page 48 ff. ref Geometric definition The four basis vectors stem from the origin of the regular tetrahedron and go to its four corners. Their coordinate addresses are 1, 0, 0, 0 , 0, 1, 0, 0 , 0, 0, 1, 0 and 0, 0, 0, 1 respectively. These may be scaled and linearly combined to span conventional XYZ space, with at least one of the four coordinates unneeded set to zero in any given quadrant. The normalization scheme is somewhat unusual in keeping all coordinates non negative. Typical of coordinate systems of this type a, a, a, a is an identity vector and may be added to normalize a result. To negate 1,0,0,0 , write &minus 1, 0, 0, 0 then add 1, 1, 1, 1 to get 0, 1, 1, 1 . Image Quadray.gif Four basis quadrays to the corners of a regular tetrahedron Pedagogical significance A typical application might set the edges of the basis tetrahedron as unit, with the quadrays considered unit on some other scale. The tetrahedron itself may also be defined as the unit of volume, although the infrastructure does not demand using this setting. The four quadrays may be linearly combined to provide integer coordinates for the inverse tetrahedron 0,1,1,1 , 1,0,1,1 , 1,1,0,1 , 1,1,1,0 , and for the cube, octahedron, rhombic dodecahedron and cuboctahedron of volumes 3, 4, 6 and 20 respectively ... Coordinates Category Coordinate systems ... more details
Paraboloidal coordinates are a three dimensional orthogonal coordinates orthogonal coordinate system math lambda, mu, nu math that generalizes the two dimensional parabolic coordinates parabolic coordinate system . Similar to the related ellipsoidal coordinates , the paraboloidal coordinate system has orthogonal coordinates orthogonal quadratic Coordinate system Coordinate surface coordinate surfaces that are not produced by rotating or projecting any two dimensional orthogonal coordinate system. Image Parabolic coordinates 3D.png thumb right 300px Coordinate system Coordinate surface Coordinate surfaces of the three dimensional paraboloidal coordinates. Basic formulae The Cartesian coordinates math x, y, z math can be produced from the ellipsoidal coordinates math lambda, mu, nu math by the equations math x 2 frac left A lambda right left A mu right left A nu right B A math math y 2 frac left B lambda right left B mu right left B nu right A B math math z frac 1 2 left A B lambda mu nu right math where the following limits apply to the coordinates math lambda B mu A nu math Consequently, surfaces of constant math lambda math are elliptic paraboloid s math frac x 2 lambda A frac y 2 lambda B 2z lambda math and surfaces of constant math nu math are likewise math frac x 2 nu A frac y 2 nu B 2z nu math whereas surfaces of constant math mu math are hyperbolic paraboloid s math frac x 2 mu A frac y 2 mu B 2z mu math Scale factors The scale factors for the paraboloidal coordinates math ... math can be expressed in the coordinates math lambda, mu, nu math by substituting the scale factors into the general formulae found in orthogonal coordinates . References reflist Bibliography cite book ... cite book author Henry Margenau Margenau H , Murphy GM year 1956 title The Mathematics of Physics ... sub k sub for sub k sub . cite book author Moon P, Spencer DE year 1988 chapter Paraboloidal Coordinates ... MathWorld description of confocal paraboloidal coordinates Orthogonal ... more details
of Mathematics CD ROM , Bipolar Coordinates , CD ROM edition 1.0, May 20, 1999 Bot generated title ref The other system is two center bipolar coordinates . There is also a third coordinate system that is based on two poles biangular coordinates . The first is based on the Apollonian circles . The curves of constant and of are circles that intersect at right angles. The coordinates have ...Cleanup date August 2007 See also two center bipolar coordinates File Bipolar isosurfaces.png thumb right 350px Bipolar coordinate system Bipolar coordinates are a two dimensional orthogonal coordinates orthogonal coordinate system . There are two commonly defined types of bipolar coordinates. ref name ... 0 and a , 0 , respectively, on the x axis of a Cartesian coordinate system . Bipolar coordinates form the basis for several sets of three dimensional orthogonal coordinates . The bipolar cylindrical coordinates are produced by projecting in the z direction. The bispherical coordinates are produced by rotating the bipolar coordinates about the math x math axis, i.e., the axis connecting the foci, whereas the toroidal coordinates are produced by rotating the bipolar coordinates about the y axis, i.e., the axis separating the foci. The classic applications of bipolar coordinates are in solving ... bipolar coordinates allow a separation of variables pde separation of variables . A typical example ... s, and Cassini oval s. However, the term bipolar coordinates is reserved for the coordinates described here, and never used to describe coordinates associated with those other curves, such as elliptic coordinates . File Bipolar coordinates.png thumb right 350px Geometric interpretation of the bipolar coordinates. The angle is formed by the two foci and the point P , whereas is the logarithm ... common definition of bipolar coordinates , is math x a frac sinh tau cosh tau cos sigma ... factors The scale factors for the bipolar coordinates , are equal math h sigma h tau frac ... more details
, related by constraint mathematics constraints on and among the coordinates. The number of dependent coordinates is the sum of the number of degrees of freedom and the number of constraints ...In the study of multibody systems, generalized coordinates are a set of coordinates used to describe .... A restriction for a set of coordinates to serve as generalized coordinates is that they should ... books.google.com books?id je0W8N5oXd4C&pg PA397 page 397 chapter 7.2.1 Selection of generalized coordinates ... coordinates are chosen to be independent of one another. The number of independent generalized coordinates is defined by the number of Degrees of freedom mechanics degrees of freedom of the system ... 2.4 Generalized coordinates publisher Springer isbn 0817642366 year 2006 ref The adjective generalized is a holdover from a period when Cartesian coordinates were the standard. Apart from practical reasons, any set of generalized coordinates is as good as another. The physics of the system is independent of the choice. However, there are more and less practical choices, that is, coordinates ... chapter 5.1 Manifolds of generalized coordinates isbn 3642053696 publisher Springer edition 5th year 2010 ref The generalized velocities are the time derivative s of the generalized coordinates of the system. Constraint equations Generalized coordinates may be independent or unconstrained , in which ... of the system. It can be advantageous to choose independent generalized coordinates, as is done ... holonomic or not, dependent generalized coordinates must be employed. Sometimes independent generalized coordinates are called internal coordinates because they are mutually independent, otherwise unconstrained ... generalized coordinates, math lbrace q 1, q 2, ..., q m rbrace math , and the time, math t math , if and only if all math m math math lbrace q j rbrace math are independent coordinates. For the system, the transformation from old coordinates to generalized coordinates may be represented as follows ... more details
coordinates blue and its projection on plane &ndash rational curve red In mathematics , homogeneous coordinates , introduced by August Ferdinand M bius in his 1827 work Der barycentrische Calc l , ref ... Main Barycentric coordinatesmathematics M bius original formulation of homogeneous coordinates ... of Modern Mathematics first David Eugene last Smith publisher J. Wiley & Sons year 1906 isbn page 53 url http books.google.com books?id 6DpBAAAAYAAJ&pg PA53 v onepage ref are a system of coordinates used in projective geometry much as Cartesian coordinate system Cartesian coordinates are used in Euclidean geometry . They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range ... coordinates of a point are multiplied by a non zero scalar then the resulting coordinates represent the same point. An additional condition must be added on the coordinates to ensure that only one set of coordinates corresponds to a given point, so the number of coordinates required is, in general ... homogeneous coordinates are required to specify a point on the projective line and three homogeneous coordinates are required to specify a point on the projective plane. This to too mathy for the lead section, try to merge with later section Therefore this system of coordinates can be explained as follows ... coordinates in V by choosing a basis, and use these in P V , the equivalence classes of proportional ... coordinates for the point nowrap x , y . Note that, since ratios are used, multiplying the three homogeneous coordinates by a common, non zero factor does not change the point represented &ndash unlike Cartesian coordinates, a single point can be represented by infinitely many homogeneous coordinates ... , y b lt . Let Z 1 t , so the coordinates of a point on the line may be written nowrap 1 a m Z , b l ... more details
Original research date June 2010 Image Hyperbolic coordinates.svg thumb 400px right Hyperbolic coordinates plotted on the Cartesian plane u in blue and v in red. In mathematics , hyperbolic coordinates are a method of locating points in Quadrant I of the Cartesian plane Why date May 2010 math x, y x 0, y 0 Q math . Hyperbolic coordinates take values in the hyperbolic plane defined as math HP u, v u in mathbb R , v 0 math . These coordinates in HP are useful for studying logarithmic comparisons of direct proportion in Q and measuring deviations from direct proportion. For math x,y math in math Q math take math u frac 1 2 ln left frac y x right math and math v sqrt xy math . Sometimes the parameter math u math is called hyperbolic angle and v the geometric mean . The inverse mapping is math x v e u , quad y v e u math . This is a continuous mapping , but not an analytic function . Quadrant model of hyperbolic geometry The correspondence math Q leftrightarrow HP math affords the hyperbolic geometry structure to Q that is erected on HP by hyperbolic motion s. The hyperbolic lines in Q are Line mathematics Ray rays from the origin or petal shaped curve s leaving and re entering the origin. The left right shift in HP corresponds to a squeeze mapping applied to Q . Note that hyperbolas in Q do not represent geodesic s in this model. If one only considers the Euclidean topology of the plane ... space HP shows that the open set Q has only the origin mathematics origin as boundary when viewed ... coordinates in the Theory of relativity see the History section below. Statistical applications Comparative ... of hyperbolic coordinates in economics Analysis of currency exchange rate fluctuation The unit currency ... rate fluctuation through hyperbolic angle provides an objective, symmetric, and consistent measure mathematics ... of the standard introductory university level textbook on relativity, hyperbolic coordinates of spacetime are called Rindler coordinates . References references David Betounes 2001 Differential ... more details
Expert subject Science date November 2008 Brinkmann coordinates named for Hans Brinkmann are a particular coordinate system for a spacetime belonging to the family of pp wave metrics . In terms of these coordinates, the metric tensor can be written as math ds 2 , H u,x,y du 2 2 du dv dx 2 dy 2 math where math partial v math , the coordinate vector field dual to the covector field math dv math , is a Minkowski space Causal structure null vector field. Indeed, geometrically speaking, it is a null geodesic congruence with vanishing optical scalars . Physically speaking, it serves as the wave vector defining the direction of wave propagation propagation for the pp wave. The coordinate vector field math partial u math can be spacelike, null, or timelike at a given Spacetime Basic concepts event in the spacetime , depending upon the sign of math H u,x,y math at that event. The coordinate vector fields math partial x , partial y math are both Minkowski space Causal structure spacelike vector fields. Each surface math u u 0 , v v 0 math can be thought of as a wavefront . In discussions of exact solutions to the Einstein field equation , many authors fail to specify the intended range of the coordinate Variable mathematics variables math u,v,x,y math . Here we should take math infty v,x,y infty, u 0 u u 1 math to allow for the possibility that the pp wave develops a null curvature singularity . References cite book author Stephani, Hans Kramer, Dietrich MacCallum, Malcolm Hoenselaers, Cornelius & Herlt, Eduard title Exact Solutions of Einstein s Field Equations location Cambridge publisher Cambridge University Press year 2003 isbn 0 521 46136 7 cite journal author H. W. Brinkmann title Einstein spaces which are mapped conformally on each other journal Math. Ann. year 1925 volume 18 pages 119 doi 10.1007 BF01208647 Category Coordinate charts in general relativity relativity stub ... more details
Image Parabolic coords.svg thumb right 384px Parabolic coordinates are a two dimensional orthogonal coordinates ... are confocal parabola s. Parabolic cylindrical coordinates A three dimensional version of parabolic coordinates is obtained by rotating the two dimensional coordinate system system about the symmetry axis of the parabolas. Parabolic coordinates have found many applications, e.g., the treatment of the Stark effect and the potential theory of the edges. Two dimensional parabolic coordinates Two dimensional parabolic coordinates math sigma, tau math are defined by the equations math x sigma tau .... Two dimensional scale factors The scale factors for the parabolic coordinates math sigma ... F math can be expressed in the coordinates math sigma, tau math by substituting the scale factors into the general formulae found in orthogonal coordinates . Three dimensional parabolic coordinates Image Parabolic coordinates 3D.png thumb right 300px Coordinate system Coordinate surface Coordinate surfaces of the three dimensional parabolic coordinates. The red paraboloid corresponds to 2, the blue ... intersect at the point P shown as a black sphere with Cartesian coordinate system Cartesian coordinates roughly 1.0, 1.732, 1.5 . The two dimensional parabolic coordinates form the basis for two sets of three dimensional orthogonal coordinates . The parabolic cylindrical coordinates are produced ... a set of confocal paraboloids, forming a coordinate system that is also known as parabolic coordinates ... F math and math nabla times mathbf F math can be expressed in the coordinates math sigma, tau, phi math by substituting the scale factors into the general formulae found in orthogonal coordinates . Bibliography ... 52 0 11515 pages 660 cite book author Henry Margenau Margenau H , Murphy GM year 1956 title The Mathematics ... author Moon P, Spencer DE year 1988 chapter Parabolic Coordinates , , title Field Theory Handbook ... of parabolic coordinates Orthogonal coordinate systems Category Coordinate systems it Coordinate ... more details
In geometry , line coordinates are used to specify the position of a Line geometry line just as point coordinates or simply Coordinate system coordinates are used to specify the position of a point. Lines ... m is the slope and b is the x intercept x intercept . This system specifies coordinates for all lines that are not vertical. However, it is more common and simpler algebraically to use coordinates nowrap ... specifies coordinates for all lines except those that pass through the origin. The geometrical interpretations .... The exclusion of lines passing through the origin can be resolved by using a system of three coordinates ... are significant, in other words if the coordinates are multiplied by a non zero scalar then line represented remains the same. So nowrap l , m , n is a system of homogeneous coordinates for the line. If points in the plane are represented by homogeneous coordinates nowrap x , y , z , the equation ... in the projective plane . The coordinates nowrap 0, 1, 0 and nowrap 1, 0, 0 represent the x and y ... a curve in the original plane determined as the Envelope mathematics envelope of the lines ... , m , n 0 represents a curve in the dual space given in homogeneous coordinates ... of curves defined as loci. Tangential equation of a point A linear equation in line coordinates ... of the point. Similarly, for a point x , y , z given in homogeneous coordinates, then the equation of the point in homogeneous tangential coordinates is lx , my , nz ... 1 & 1 l 2 & m 2 & 1 l 3 & m 3 & 1 end vmatrix 0. math For homogeneous coordinates, the intersection ... math begin vmatrix l 1 & m 1 & n 1 l 2 & m 2 & n 2 l 3 & m 3 & n 3 end vmatrix 0. math Dually, the coordinates ... Lines in three dimensional space main Pl cker coordinates For two given points in the plane, x sub ... of homogeneous line coordinates in three dimensional space called Pl cker coordinates . Six numbers in a set of coordinates only represent a line when they satisfy an additional equation. This system ... more details
In mathematics , orthogonal coordinates are defined as a set of d coordinates q q sup 1 sup , q sup 2 ... operators in the new coordinates, e.g., the gradient , the Vector Laplacian Laplacian , the divergence and the Curl mathematics curl . A simple method for generating orthogonal coordinates ... k sub is a constant. For example, the three dimensional Cartesian coordinate system Cartesian coordinates .... Orthogonal coordinates are a special but extremely common case of curvilinear coordinates . While vector operations and physical laws are normally easiest to derive in Cartesian coordinates , non Cartesian orthogonal coordinates are often used instead for the solution of various problems, especially ... coordinates is that they can be chosen to match the symmetry of the problem. For example, the pressure ... coordinates, however the pressure predominantly moves away from the center, so that in spherical coordinates the problem becomes very nearly one dimensional since the pressure wave dominantly depends ... pipe in Cartesian coordinates, one has to solve a difficult two dimensional boundary value problem involving a partial differential equation, but in cylindrical coordinates the problem becomes ... to prefer orthogonal coordinates instead of general curvilinear coordinates is simplicity many complications arise when coordinates are not orthogonal. For example, in orthogonal coordinates ... 523, 655 666. ref . Orthogonal coordinates never have off diagonal terms in their metric tensor ... coordinates x , y . A complex number z x iy can be formed from the real coordinates x and y , where ... coordinates in three and higher dimensions can be generated from an orthogonal two dimensional coordinate system, either by projecting it into a new dimension cylindrical coordinates or by rotating ... system, such as the ellipsoidal coordinates . More general orthogonal coordinates may be obtained ... . Basis vectors In Cartesian coordinates , the basis vectors are fixed constant . In the more ... more details
math h r 1 math , as in spherical coordinates . The scale factors for the two conical coordinates ... 659 cite book author Henry Margenau Margenau H , Murphy GM year 1956 title The Mathematics of Physics ... pages 118 119 id ASIN B000MBRNX4 cite book author Moon P, Spencer DE year 1988 chapter Conical Coordinates ... MathWorld description of conical coordinates Orthogonal coordinate systems Category Coordinate ... more details
In Riemannian geometry , a branch of mathematics , harmonic coordinates are a coordinate system nowrap x sup 1 sup ,..., x sup n sup on a Riemannian manifold each of whose coordinate functions x sup i sup is harmonic function harmonic , meaning that it satisfies Laplace s equation math Delta x i 0. , math Here &Delta is the Laplace Beltrami operator . Equivalently, regarding a coordinate system as a local diffeomorphism nowrap &phi M &rarr R sup n sup , the coordinate system is harmonic if and only if &phi is a harmonic map of Riemannian manifolds, roughly meaning that it minimizes the elastic energy of stretching M into R sup n sup . The elastic energy is expressed via the Dirichlet energy functional math E varphi int M d varphi 2 ,dV. math In two dimensions, harmonic coordinates have been well understood for more than a century, and are closely related to isothermal coordinates , the latter being a special case of the former. Harmonic coordinates in higher dimensions were developed initially in the context of general relativity by harvtxt Einstein 1916 see harmonic coordinate condition . They were then introduced into Riemannian geometry by harvtxt Sabitov efel 1976 and later were studied by harvtxt DeTurck Kazdan 1981 . The essential motivation for introducing harmonic coordinate systems is that the metric tensor is especially smooth function smooth when written in these coordinate systems. Harmonic coordinates are characterized in terms of the Christoffel symbols by means of the relation math g ij Gamma ij k 0 , math and indeed, for any coordinate system at all, math Delta x k g ij Gamma ij k. math Harmonic coordinates always exist locally , a result which follows easily .... The basic regularity theorem concerning the metric in harmonic coordinates is that if the components ..., then they are in that same H lder space when expressed in harmonic coordinates. In general relativity , harmonic coordinates are solutions of the wave equation instead of the Laplace . This is known ... more details
is not the same as the centroid the centroid has Barycentric coordinatesmathematics barycentric coordinates ...In geometry , the trilinear coordinates of a point relative to a given triangle describe the relative distances from the three sides of the triangle. Trilinear coordinates are an example of homogeneous coordinates . They are often called simply trilinears . Examples The incenter has trilinears 1 1 1 that is, the directed distances from the incenter to the sidelines BC , CA , AB of a triangle ABC are proportional to the actual distances, which are the ordered triple r , r , r , where r is the inradius of triangle ABC . Note that the notation x y z using colons distinguishes trilinears from actual directed distances, kx , ky , kz , which is the usual notation for an ordered triple, and which may be obtained from x y z using the number math k frac 2 sigma ax by cz math where a , b , c are the respective sidelengths BC , CA , AB , and area of ABC . Comma notation for trilinears should be avoided ... has barycentric coordinates a b c where a , b , c are the sidelengths of the triangle. Conversely ... between trilinear coordinates and 2D Cartesian coordinates . Given a reference triangle ABC express the position of the vertex B in terms of an ordered pair of Cartesian coordinates and represent ... trilinear coordinates x y z then the conversion formulas are as follows math x y z frac beta a frac ... . math If an arbitrary origin is chosen where the Cartesian coordinates of the vertices are known and represented by the vectors u A u , u B u and u C u and if the point P has trilinear coordinates x y z , then the Cartesian coordinates of u P u are the weighted average of the Cartesian coordinates of these vertices using the barycentric coordinates ax , by and cz as the weights. Hence math underline ... MathWorld title Trilinear Coordinates urlname TrilinearCoordinates External links http faculty.evansville.edu ... coordinates and barycentric for more than 3200 triangle centers Category Linear algebra Category ... more details
In mathematics , specifically in differential geometry , isothermal coordinates on a Riemannian manifold are local coordinates where the metric tensor metric is conformal to the Euclidean metric . This means that in isothermal coordinates, the Riemannian metric locally has the form math g e varphi dx 1 2 cdots dx n 2 , math where math varphi math is a smooth function . Isothermal coordinates on surfaces were first introduced by Carl Friedrich Gauss Gauss . Korn and Lichtenstein proved that isothermal coordinates exist around any point on a two dimensional Riemannian manifold. On higher dimensional Riemannian manifolds a necessary and sufficient condition for their local existence is the vanishing of the Weyl tensor and of the Cotton tensor . Isothermal coordinates on surfaces The first result on the existence of isothermal coordinates was due to harvtxt Korn 1916 and harvtxt Lichtenstein 1916 . Later accounts were given by harvtxt Morrey 1938 , harvtxt Ahlfors 1955 , harvtxt Bers 1952 and harvtxt Chern 1955 . A particularly simple account using the Hodge star operator is given in harvtxt DeTurck Kazdan 1981 . Beltrami equation The existence of isothermal coordinates can be proved ref harvnb Imayoshi Taniguchi 1992 pp 20&ndash 21 ref by applying known existence theorems for the Beltrami equation , which rely on L sup p sup estimates for singular integral operator s of Alberto Calderon Calderon and Antoni Zygmund Zygmund . ref harvnb Ahlfors 1966 pp 85&ndash 115 ref ref harvnb Imayoshi Taniguchi 1992 pp 92&ndash 104 ref A simpler approach to the Beltrami equation has been given ... coordinates u , v the metric should take the form math ds 2 rho du 2 dv 2 math with 0 smooth ... over w z , d overline z 2, math so that the coordinates u , v will be isothermal if the Beltrami equation ... star operator New coordinates u and v are isothermal provided that math star du dv, math where math ... isothermal coordinates. Gaussian curvature In the isothermal coordinates u , v , the Gaussian curvature ... more details
Ellipsoidal coordinates are a three dimensional orthogonal coordinates orthogonal coordinate system math lambda, mu, nu math that generalizes the two dimensional elliptic coordinates elliptic coordinate system . Unlike most three dimensional orthogonal coordinates orthogonal coordinate system s that feature quadratic Coordinate system Coordinate surface coordinate surfaces , the ellipsoidal coordinate system is not produced by rotating or projecting any two dimensional orthogonal coordinate system. Basic formulae The Cartesian coordinates math x, y, z math can be produced from the ellipsoidal coordinates math lambda, mu, nu math by the equations math x 2 frac left a 2 lambda right left a 2 mu right left a 2 nu right left a 2 b 2 right left a 2 c 2 right math math y 2 frac left b 2 lambda right left b 2 mu right left b 2 nu right left b 2 a 2 right left b 2 c 2 right math math z 2 frac left c 2 lambda right left c 2 mu right left c 2 nu right left c 2 b 2 right left c 2 a 2 right math where the following limits apply to the coordinates math lambda c 2 mu b 2 nu a 2 . math Consequently, surfaces of constant math lambda math are ellipsoid s math frac x 2 a 2 lambda frac y 2 b 2 lambda frac z 2 c 2 lambda 1, math whereas surfaces of constant math mu math are hyperboloid s of one sheet math frac x 2 a 2 mu frac y 2 b 2 mu frac z 2 c 2 mu 1, math because the last term in the lhs is negative ... mathbf F math can be expressed in the coordinates math lambda, mu, nu math by substituting the scale factors into the general formulae found in orthogonal coordinates . See also Focaloid shell given ... author Margenau H, Murphy GM year 1956 title The Mathematics of Physics and Chemistry publisher ... DE year 1988 chapter Ellipsoidal Coordinates , , title Field Theory Handbook, Including Coordinate ... Press location New York isbn 978 0750626347 pages 19&ndash 29 Uses , , coordinates that have ... MathWorld description of confocal ellipsoidal coordinates Orthogonal coordinate systems ... more details
Image Toroidal coordinates.png thumb 350px right Illustration of toroidal coordinates, which are obtained by rotating a two dimensional bipolar coordinates bipolar coordinate system about the axis separating ..., at Cartesian coordinates roughly 0.996, &minus 1.725, 1.911 . Toroidal coordinates are a three dimensional orthogonal coordinates orthogonal coordinate system that results from rotating the two dimensional bipolar coordinates bipolar coordinate system about the axis that separates its two foci. Thus, the two Focus geometry foci math F 1 math and math F 2 math in bipolar coordinates become ... The most common definition of toroidal coordinates math sigma, tau, phi math is math x a frac sinh ... Rotating this two dimensional bipolar coordinates bipolar coordinate system about the vertical axis ... math tori are centered in the math xy math plane. Inverse transformation The , , coordinates may be calculated from the Cartesian coordinates x , y , z as follows. The azimuthal angle is given ... interpretation of the coordinates and of a point P . Observed in the plane of constant azimuthal angle , toroidal coordinates are equivalent to bipolar coordinates . The angle is formed by the two ... the x y plane. Scale factors The scale factors for the toroidal coordinates math sigma math and math ... F math and math nabla times mathbf F math can be expressed in the coordinates math sigma, tau, phi math by substituting the scale factors into the general formulae found in orthogonal coordinates ... solution via separation of variables in toroidal coordinates. Making the substitution math V U ... for associated Legendre functions. The classic applications of toroidal coordinates are in solving partial differential equations , e.g., Laplace s equation for which toroidal coordinates allow a separation of variables or the Helmholtz equation , for which toroidal coordinates do not allow a separation ... year 2006 title Alternative separation of Laplace s equation in toroidal coordinates and its application ... more details
affine span , and Cartesian coordinates in two dimensional space Curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may ... coordinate system to its curvilinear coordinates and back. The name curvilinear coordinates , coined ... Cartesian coordinate s, we can represent a point in space by the coordinates math x 1, x 2 math and in vector ... 2 math are basis vector s. We can describe the same point in curvilinear coordinates in a similar manner, except that the coordinates are now math xi 1, xi 2 math and the position vector is math mathbf ... examples of curvilinear systems are Cylindrical coordinate system cylindrical and spherical coordinates spherical polar coordinates for R sup 3 sup . While a Cartesian Coordinate system Coordinate ... in spherical polar coordinates is the surface of a unit sphere in R sup 3 sup &mdash which obviously is curved. Coordinates are often used to define the location or distribution of physical quantities which may be scalar mathematics scalar s, vector geometric vector s, or tensor s. Depending on the application ... sup e.g., motion in the field of a point mass charge , is usually easier to solve in spherical coordinates spherical polar coordinates than in Cartesian coordinates. Also boundary conditions may enforce symmetry. One would describe the motion of a particle in a rectangular box in Cartesian coordinates, whereas one would prefer spherical coordinates for a particle in a sphere. Many of the concepts in vector calculus , which are given in Cartesian or spherical coordinates spherical polar coordinates, can be formulated in arbitrary curvilinear coordinates. This gives a certain economy of thought ... such as the gradient , divergence , curl , and the Laplacian . Curvilinear Coordinates from ... . General curvilinear coordinates Image General curvilinear coordinates 1.svg thumb left 350px Fig. 1 Coordinate surfaces, coordinate lines, and coordinate axes of general curvilinear coordinates. In Cartesian ... more details
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