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Topic: Curvilinear coordinates

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	Coordinates (mathematics) - Wikipedia, the free encyclopedia
		The coordinates of a point are the components of a tuple of numbers used to represent the location of the point in the plane or space.
		The polar coordinate systems are coordinate systems in which a point is identified by a distance from some fixed feature in space and one or more subtended angles.
		In terms of the Cartesian coordinate system, one usually picks O to be the origin (0,0) and L to be the positive x-axis (the right half of the x-axis).
	en.wikipedia.org /wiki/Coordinates_(elementary_mathematics) (1186 words)

	Curvilinear coordinates (Site not responding. Last check: 2007-10-21)
		Curvilinear coordinates are a coordinatesystem based on some transformation of the standard coordinate system, such that the transformation is locally invertible ateach point.
		This is useful in that working with the curvilinear coordinate system in one situation may be more simple than in the Cartesian coordinate system.This also has consequences thatwe can express many of the concepts in vector calculus which aregiven in Cartesian or spherical coordinates or any otherarbitrary coordinate system, also in curvilinear coordinates.
		Since we use curvilinear coordinates to aid in the calculation in vector calculus, there are adjustments we need to make in the calculation of line, surface and volumeintegrals.
	www.therfcc.org /curvilinear-coordinates-156300.html (331 words)

	Boundary fitted coordinates
		The basic idea of a boundary-conforming curvilinear coordinate system is to have some coordinate line (in 2D, surface in 3D) coincident with each boundary segment, analogous to the way in which lines of constant radial coordinate coincide with circles in the cylindrical coordinate system.
		The other curvilinear coordinate, analogous to the angular coordinate in the cylindrical system, will vary along the boundary segment and clearly must do so monotonically, else the same pair of values of the curvilinear coordinates will occur at two different physical points.
		There must, or course, be a unique correspondence between the Cartesian (or other basis system) and the curvilinear coordinates, i.e., the mapping of the physical region onto the transformed region must be one-to-one, so that every point in the physical field corresponds to one, and only one, point in the transformed field, and vice versa.
	www.cse.ucsc.edu /~shreyas/btp/node14.html (442 words)

	BOUNDARY-CONFORMING COORDINATE SYSTEMS
		With values of the cartesian coordinates known in the field as functions of the curvilinear coordinates, the network of intersecting lines formed by contours (surfaces in 3D) on which a curvilinear coordinate is constant, i.e., the curvilinear coordinate system, provides the needed organization of the discretization with conformation to the physical boundary.
		The cartesian coordinates of the desired points on the physical boundary 2-9 are to be used on the slit in the generation of the grid to the left of the slit in the transformed region, while those on the physical boundary 5-9 are used for generation to the right of the slit.
		In all cases the interpretation of the positive directions of the curvilinear coordinates must be such as to preserve the direction in the physical region, i.e., on the same sheet, as the cut is crossed.
	www.erc.msstate.edu /publications/gridbook/chap02/text.html (12591 words)

	III. TRANSFORMATION RELATIONS
		Partial derivatives with respect to cartesian coordinates are related to partial derivatives with respect to curvilinear coordinates by the chain rule which may be written in either of two ways.
		The curvilinear coordinate lines of a three-dimensional system are space curves formed by the intersection of surfaces on which one coordinate is constant.
		in the expressions are the derivatives of the curvilinear coordinates with respect to the cartesian coordinates, and this notation, rather than the cross-products, often appears in the literature.
	www.erc.msstate.edu /publications/gridbook/chap03/text.html (2615 words)

	PlanetMath: gradient in curvilinear coordinates (Site not responding. Last check: 2007-10-21)
		We give the formulas for the gradient expressed in various curvilinear coordinate systems.
		"gradient in curvilinear coordinates" is owned by stevecheng.
		This is version 2 of gradient in curvilinear coordinates, born on 2005-08-11, modified 2005-08-11.
	planetmath.org /encyclopedia/GradientInCurvilinearCoordinates.html (198 words)

	Curvilinear Coordinates Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-10-21)
		Curvilinear coordinates are a coordinate system based on some transformation of the standard coordinate system.
		Depending on the application, a curvilinear coordinate system may be simpler than the Cartesian coordinate system.
		This also has consequences that we can express many of the concepts in vector calculus which are given in Cartesian or spherical coordinates or any other arbitrary coordinate system, also in curvilinear coordinates.
	www.karr.net /search/encyclopedia/Curvilinear_coordinates (566 words)

	Encyclopedia: Curvilinear coordinates (Site not responding. Last check: 2007-10-21)
		In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.
		But from the definition of the curvilinear coordinates, In calculus, the chain rule is a formula for the derivative of the composition of two functions.
		In curvilinear coordinates, we can express the divergence, curl and gradient of a function or vector field as follows: In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point.
	www.nationmaster.com /encyclopedia/Curvilinear-coordinates (631 words)

	The Waveguide Mesh in General Curvilinear Coordinates
		A generalization of the waveguide mesh to arbitrary curvilinear coordinates is useful in that it becomes possible to model boundary conditions which may not be simply aligned with a rectilinear grid.
		The admittances at the parallel junctions are defined in a manner similar to those of the DWN in rectilinear coordinates.
		FDTD in general curvilinear coordinates has developed in a similar way; most formulations are slightly different in that they are based on a tensor density formulation [
	ccrma-www.stanford.edu /~bilbao/master/node131.html (756 words)

	WL :: generalisation of shallow water equations to spherical curvilinear coordinates
		Also, the application of spherical coordinates as defined by the World Geodetic System (WGS84) reduces the number of coordinate transformations in areas that are covered by more than one local coordinate system.
		In the current project, shallow water equations have been derived for spherical curvilinear coordinates in order to investigate the feasibility of the application of these coordinates in the Delft3D-FLOW solver.
		It appears that the equations in spherical curvilinear coordinates are identical to those equations which had already been implemented for Cartesian curvilinear coordinates.
	www.wldelft.nl /rnd/intro/topic/2003-swe (318 words)

	Re: Orthogonal curvilinear coordinates
		The area, in which I want to generate orthogonal coordinates, is a 2x2 square with the upper line replaced by a cos-function (y=1.6-0.4cos(pix), x=-1..1).
		Re: Orthogonal curvilinear coordinates - andy, Thu, 3 Dec 1998, 5:25 a.m.
		Re: Orthogonal curvilinear coordinates - andy, Thu, 3 Dec 1998, 8:24 a.m.
	www.cfd-online.com /Forum/main_archive_1998.cgi?read=1542 (240 words)

	The Need for Ortho-Curvilinear Coordinates (Site not responding. Last check: 2007-10-21)
		We shall restrict ourselves to ortho-curvilinear coordinate systems, that is: coordinate systems in which the coordinate lines intersect perpendicularly.
		Because the boundary curve is not co-aligned with the coordinate lines and does not go through the cell centres, where the quantities of interest are typically defined, this is a complex operation.
		For general geometries standard coordinate systems cannot be used to obtain boundary-fitted grids, and general ortho-curvilinear coordinates must be used.
	plasimo.phys.tue.nl /generated-docs/plgrid/html/Need_Ortho_Curvilinear_Coor.html (1353 words)

	International Journal of Mechanical Engineering Education (Site not responding. Last check: 2007-10-21)
		A systematic method to generate expressions appearing in physics and engineering and valid in curvilinear orthogonal coordinates is presented.
		The method, which is called 'the method of local cartesian coordinates' achieves the same as 'the method of moving axes' presented in Love but with a smaller effort and with more familiar mathematical tools.
		For this purpose, polar coordinates are employed extensively as a demonstration example.
	www.me.umist.ac.uk /ijmee/26-4p5.htm (125 words)

	Curvilinear coordinates -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		Curvilinear coordinates are a (A system that uses coordinates to establish position) coordinate system based on some transformation of the standard coordinate system, such that the transformation is locally invertible at each point.
		Since we use curvilinear coordinates to aid in the calculation in (Click link for more info and facts about vector calculus) vector calculus, there are adjustments we need to make in the calculation of line, surface and volume integrals.
		Likewise, if we are interested in a (Click link for more info and facts about surface integral) surface integral, the relevant calculation, with the parametrisation of the surface in Cartesian coordinates is:
	www.absoluteastronomy.com /encyclopedia/c/cu/curvilinear_coordinates.htm (350 words)

	2.10.5 Horizontal grid
		The horizontal coordinate system is orthogonal curvilinear meaning we can not necessarily treat the two horizontal directions as separable.
		All that is necessary to distinguish between the coordinate systems is to initialize the grid data (descriptors) appropriately.
		Curvilinear coordinates are selected when the logical flag usingCurvilinearGrid in namelist PARM04 is set to true.
	mitgcm.org /~adcroft/l2h/node47.html (746 words)

	Maybe this Explains the Economic Cycle... best Curvilinear Coordinates (Site not responding. Last check: 2007-10-21)
		Multiinstantons in curvilinear coordinates Multiinstantons in curvilinear coordinates The 'tHooft's 5N-parametric multiinstanton solution is generalized to curvilinear coordinates.
		Multiinstantons in curvilinear coordinates Multiinstantons in curvilinear coordinates Comment: LaTeX file, 17 pages The 'tHooft's 5N-parametric multiinstanton solution is generalized to curvilinear coordinates.
		Instantons in curvilinear coordinates Instantons in curvilinear coordinates The multi-instanton solutions by 'tHooft and Jackiw, Nohl and Rebbi are generalized to curvilinear coordinates.
	ascot.pl /th/Fourier3/Curvilinear-Coordinates.htm (418 words)

	9.4.1 Some rules of tensor analysis on manifolds
		However, to facilitate the eventual transition to curvilinear coordinates, the approach taken here is to employ the notation of curvilinear tensor analysis (e.g., Aris 1962).
		The metric for spherical coordinates on a sphere is diagonal.
		This is a trivial property for Cartesian coordinates on a plane, in which case the metric is the constant unit tensor and the covariant derivative a partial derivative
	www.gfdl.noaa.gov /~smg/MOM/web/guide_parent/s2node107.html (738 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		A field with curvilinear coordinates consists of a regular array of grid cells.
		The coordinates of the grid nodes are specified explicitly.
		The coordinate type of the field is specified in the parameter section of the AmiraMesh file.
	www.amiravis.com /usersguide31/amiramesh/AmiraMeshCurvilinear.html (120 words)

	BldProp: beam property definition (Site not responding. Last check: 2007-10-21)
		The table entries are mapped to physical locations along the beam span by letting the curvilinear variable describing length along the curve defining the geometry of the beam match the curvilinear variable
		are the coordinates of the shear center which are used to transform the sectional torsional and shearing stiffnesses to the corresponding stiffnesses in the local axis system
		are the coordinates of the centroid which are used to transform the sectional axial and bending stiffnesses to the corresponding stiffnesses in the local axis system.
	www.ae.gatech.edu /people/obauchau/manual/BldProp.html (783 words)

	3 Using the Lattice Data Type
		The lattices on the module input port and on the module output port use the same coordinate values, which are passed directly from input to output, but the data values change from input to output as they are processed by the user function.
		Coordinates for perimeter lattices are stored in row-major format in the C convention (see Figure 3-10).
		Curvilinear lattices are used to store datasets where the data values at a node need to be associated explicitly with the Cartesian coordinates of the node.
	www.nag.co.uk /visual/IE/iecbb/DOC/html/unix-mwg-chap03.htm (3369 words)

	No Title
		Ability to plot data on general curvilinear grids: Since four-dimensional data cannot be directly displayed on a flat computer screen, it is necessary to reduce the dimensionality of the data before it is displayed.
		Data slicing and averaging may be done with respect to a fixed physical coordinate even if the data is stored in a curvilinear coordinate system.
		Extensibility via subprograms supplied by the user at run time: Arbitrarily complex curvilinear coordinates may be accommodated through user-supplied subprograms that convert between physical space coordinates and the array indices for gridded data on a curvilinear mesh.
	www.atmos.washington.edu /ive/ive_overview/ive_overview.html (892 words)

	Parametric-Equation Representative of Biconcave Erythrocytes -- from Mathematica Information Center
		The representation of the shape of a biconcave erythrocyte by a set of three parametric equations was achieved by using the expressions that transform the curvilinear coordinates from the disc-cyclide coordinate system [denoted J2R; Moon and Spencer (1998), Field Theory Handbook, Springer-Verlag, Berlin] to Cartesian coordinates.
		The equations are products of elliptic functions, so the challenge was to relate the three major 'shape-defining' measurements of the human erythrocyte in Cartesian coordinates to three parameters in the new curvilinear coordinates, to give a realistic representation of the shape of the membrane-surface.
		The relationships between the coefficients of the Cartesian degree-4 surface that describes the discocyte and the coordinate transformation equations were derived with the aid of Mathematica; and this 'package'.
	library.wolfram.com /infocenter/Articles/1415 (222 words)

	Re: curvilinear coordinates
		Re: curvilinear coordinates - sibel, Tue, 1 Feb 2005, 12:41 a.m.
		Re: curvilinear coordinates - zxaar, Tue, 1 Feb 2005, 12:52 a.m.
		Re: curvilinear coordinates - sibel, Tue, 1 Feb 2005, 2:33 a.m.
	www.cfd-online.com /Forum/main_archive.cgi?read=35097 (143 words)

	[No title]
		In this case the coordinate type must be checked dynamically and the pointer to the coordinate object has to be down-cast explicitly before it can be used.
		Same as for uniform or stacked coordinates, in the case of rectilinear coordinates the grid cells are aligned to the axes, but the grid spacing may vary from cell to cell in each direction.
		In the case of curvilinear coordinates, the position of each grid node is stored explicitly as a 3D vector of floats.
	www.tgs.com /support/amira_doc/Documentation/PROGGUIDE/regcoords.html (528 words)

	Re: curvilinear coordinates
		curvilinear coordinates - Sibel, Mon, 31 Jan 2005, 4:47 p.m.
		Re: curvilinear coordinates - Runge_Kutta, Mon, 31 Jan 2005, 8:01 p.m.
		Re: curvilinear coordinates - sibel, Tue, 1 Feb 2005, 1:53 a.m.
	www.cfd-online.com /Forum/main_archive.cgi?read=35094 (192 words)

	Ferret FAQ (Site not responding. Last check: 2007-10-21)
		The analytical methods involve regridding the sigma coordinates to a grid based on a true depth axis.
		The results of this method can then be combined in calculations with other depth-gridded fields (The same procedures apply to pressure-gridded fields.) The graphical methods are "faithful" in the sense that they do not involve regridding the data to another grid.
		coordinate on the tdepth axis (r0 is a 4-dimensional variable with !
	ferret.wrc.noaa.gov /Ferret/FAQ/data_management/sigma_coordinate_demo.html (834 words)

	Old Meetings: RHT95
		With respect to the application of curvilinear, bodyfitted grids, special emphasis is placed on the various effects encountered when a non-orthogonal grid is used.
		A general expression for the pathlength derivative of radiation intensity is given for spatial directional orthogonal curvilinear coordinates system.
		The problem is solved for a set of curvilinear coordinates (s,n) moving along the inner wall.
	www.ichmt.org /abstracts/RHT-95/session-2.html (2378 words)

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