query: who is credited with developing the xy coordinate plane

pos: Cartesian coordinate system the origin. One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, "n" Cartesian coordinates (an element of real "n"-space) specify the point in an "n"-dimensional Euclidean space for any dimension "n". These coordinates are equal, up to sign, to distances from the point to "n" mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: "Cartesius") revolutionized mathematics by providing the

neg: Cartesian coordinate system leave the orientation unchanged. Once the "x"- and "y"-axes are specified, they determine the line along which the "z"-axis should lie, but there are two possible orientation for this line. The two possible coordinate systems which result are called 'right-handed' and 'left-handed'. The standard orientation, where the "xy"-plane is horizontal and the "z"-axis points up (and the "x"- and the "y"-axis form a positively oriented two-dimensional coordinate system in the "xy"-plane if observed from "above" the "xy"-plane) is called right-handed or positive. The name derives from the right-hand rule. If the index finger of the right hand is pointed forward,

neg: Operation of computed tomography the figure. The "xy" and "ts" coordinate systems share the origin (6) and they are positioned on the same plane. That is, the "xy" plane and the "ts" plane are the same plane. Henceforth, this virtual plane will be called “the datum plane”. In addition, a virtual circle centered at the abovementioned origin (6) is set on the datum plane (it will be called “the datum circle” henceforth). This datum circle (6) will be represents the orbit of the parallel beam irradiation optical system. Naturally, the origin (6), the datum circle (5), and the datum coordinate systems are virtual features

neg: State Plane Coordinate System coordinate system to specify locations rather than a more complex spherical coordinate system (the geographic coordinate system of latitude and longitude). By using the Cartesian coordinate system's simple XY coordinates, "plane surveying" methods can be used, speeding up and simplifying calculations. Second, the system is highly accurate within each zone (error less than 1:10,000). Outside a specific state plane zone accuracy rapidly declines, thus the system is not useful for regional or national mapping. Most state plane zones are based on either a transverse Mercator projection or a Lambert conformal conic projection. The choice between the two map projections is

neg: Cupola (geometry) base (or the side opposite the base, which can be called the top) to be a regular polygon, but it is convenient to consider the case where the cupola has its maximal symmetry, C. In that case, the top is a regular "n"-gon, while the base is either a regular 2"n"-gon or a 2"n"-gon which has two different side lengths alternating and the same angles as a regular 2"n"-gon. It is convenient to fix the coordinate system so that the base lies in the "xy"-plane, with the top in a plane parallel to the "xy"-plane. The "z"-axis is the "n"-fold

neg: Vector notation that define "r" and "θ" explicitly. This can be unwieldy, but is useful for avoiding the confusion with two-dimensional rectangular vectors that arises from using ordered pair or matrix notation. A two-dimensional vector whose magnitude is 5 units and whose direction is "π"/9 radians (20°) can be specified using either of the following forms: A cylindrical vector is an extension of the concept of polar vectors into three dimensions. It is akin to an arrow in the cylindrical coordinate system. A cylindrical vector is specified by a distance in the "xy"-plane, an angle, and a distance from the "xy"-plane (a

neg: Cartesian coordinate system is also meant to point "towards" the observer, whereas the "middle"-axis is meant to point "away" from the observer. The red circle is "parallel" to the horizontal "xy"-plane and indicates rotation from the "x"-axis to the "y"-axis (in both cases). Hence the red arrow passes "in front of" the "z"-axis. Figure 8 is another attempt at depicting a right-handed coordinate system. Again, there is an ambiguity caused by projecting the three-dimensional coordinate system into the plane. Many observers see Figure 8 as "flipping in and out" between a convex cube and a concave "corner". This corresponds to the two possible

neg: Proper time the proper time equation is needed, along with a parameterized description of the path being taken, as shown below. Let there be an observer "C" on a disk rotating in the "xy" plane at a coordinate angular rate of formula_15 and who is at a distance of "r" from the center of the disk with the center of the disk at "x"="y"="z"=0. The path of observer "C" is given by formula_16, where formula_17 is the current coordinate time. When "r" and formula_15 are constant, formula_19 and formula_20. The incremental proper time formula then becomes So for an observer rotating at

neg: Cartesian coordinate system and the unit points on the three axes are , , and . There are no standard names for the coordinates in the three axes (however, the terms "abscissa", "ordinate" and "applicate" are sometimes used). The coordinates are often denoted by the letters "X", "Y", and "Z", or "x", "y", and "z". The axes may then be referred to as the "X"-axis, "Y"-axis, and "Z"-axis, respectively. Then the coordinate planes can be referred to as the "XY"-plane, "YZ"-plane, and "XZ"-plane. In mathematics, physics, and engineering contexts, the first two axes are often defined or depicted as horizontal, with the third

neg: Hyperbolic geometry the xy-plane. This model is not as widely used as other models but nevertheless is quite useful in the understanding of hyperbolic geometry. The band model employs a portion of the Euclidean plane between two parallel lines. Distance is preserved along one line through the middle of the band. Assuming the band is given by formula_23, the metric is given by formula_24. All models essentially describe the same structure. The difference between them is that they represent different coordinate charts laid down on the same metric space, namely the hyperbolic plane. The characteristic feature of the hyperbolic plane itself is

neg: Cartesian coordinate system before cementing the "x"-, "y"-, and "z"-axis concepts, by starting with 2D mnemonics (e.g. 'Walk along the hall then up the stairs' akin to straight across the "x"-axis then up vertically along the "y"-axis). Computer graphics and image processing, however, often use a coordinate system with the "y"-axis oriented downwards on the computer display. This convention developed in the 1960s (or earlier) from the way that images were originally stored in display buffers. For three-dimensional systems, a convention is to portray the "xy"-plane horizontally, with the "z"-axis added to represent height (positive up). Furthermore, there is a convention to orient

neg: Function of several real variables example can be extended easily to higher dimensions: for non-zero real constants , which describes a -dimensional hyperplane. The Euclidean norm: is also a function of "n" variables which is everywhere defined, while is defined only for . For a non-linear example function in two variables: which takes in all points in , a disk of radius "punctured" at the origin in the plane , and returns a point in . The function does not include the origin , if it did then would be ill-defined at that point. Using a 3d Cartesian coordinate system with the xy plane as

neg: Function of several real variables with base area and height measured perpendicularly from the base. The domain restricts all variables to be positive since lengths and areas must be positive. For an example of a function in two variables: where and are real non-zero constants. Using the three-dimensional Cartesian coordinate system, where the xy plane is the domain and the z axis is the codomain , one can visualize the image to be a two-dimensional plane, with a slope of in the positive x direction and a slope of in the positive y direction. The function is well-defined at all points in . The previous

neg: Parallel coordinates Each Census, 1790-1890" in 1898. They were popularised again 79 years later by Alfred Inselberg in 1959 and systematically developed as a coordinate system starting from 1977. Some important applications are in collision avoidance algorithms for air traffic control (1987—3 USA patents), data mining (USA patent), computer vision (USA patent), Optimization, process control, more recently in intrusion detection and elsewhere. On the plane with an xy cartesian coordinate system, adding more dimensions in parallel coordinates (often abbreviated ||-coords or PCP) involves adding more axes. The value of parallel coordinates is that certain geometrical properties in high dimensions transform into easily

neg: Pulsed electron paramagnetic resonance in the xy-plane, as B does. The motion of the magnetization now defines a cone. That means as Δω becomes larger, the magnetization is tipped less effectively into the xy-plane, and the FID signal decreases. In broad EPR spectra where Δω > ω it is not possible to tip all the magnetization into the xy-plane to generate a strong FID signal. This is why it is important to maximize ω or minimize the π/2 pulse length for broad EPR signals. So far the magnetization was tipped into the xy-plane and it remained there with the same magnitude. However, in reality

neg: Winding number the "xy" plane. By translating the coordinate system, we can extend this definition to include winding numbers around any point "p". Winding number is often defined in different ways in various parts of mathematics. All of the definitions below are equivalent to the one given above: A simple combinatorial rule for defining the winding number was proposed by August Ferdinand Möbius in 1865 and again independently by James Waddell Alexander II in 1928. Any curve partitions the plane into several connected regions, one of which is unbounded. The winding numbers of the curve around two points in the same region

neg: Bohr model Wilson–Sommerfeld quantization condition where "p" is the radial momentum canonically conjugate to the coordinate "q" which is the radial position and "T" is one full orbital period. The integral is the action of action-angle coordinates. This condition, suggested by the correspondence principle, is the only one possible, since the quantum numbers are adiabatic invariants. The Bohr–Sommerfeld model was fundamentally inconsistent and led to many paradoxes. The magnetic quantum number measured the tilt of the orbital plane relative to the "xy"-plane, and it could only take a few discrete values. This contradicted the obvious fact that an atom could be turned

neg: Rigid body is given by where Q is the point fixed in B that instantaneously coincident with R at the instant of interest. This equation is often combined with Acceleration of two points fixed on a rigid body. If "C" is the origin of a local coordinate system "L", attached to the body, where In 2D, the angular velocity is a scalar, and matrix A(t) simply represents a rotation in the "xy"-plane by an angle which is the integral of the angular velocity over time. Vehicles, walking people, etc., usually rotate according to changes in the direction of the velocity: they move

neg: Leibniz integral rule derivative with respect to a coordinate system attached to the moving surface. Having found the derivative, variables can be switched back to the original frame of reference. We notice that (see article on curl) and that Stokes theorem equates the surface integral of the curl over Σ with a line integral over ∂Σ: The sign of the line integral is based on the right-hand rule for the choice of direction of line element "ds. To establish this sign, for example, suppose the field F points in the positive "z"-direction, and the surface Σ is a portion of the "xy"-plane with

neg: Hyperbolic geometry the Euclidean plane, bounded by a line "B" of the plane, to be a model of the hyperbolic plane. The line "B" is not included in the model. The Euclidean plane may be taken to be a plane with the Cartesian coordinate system and the x-axis is taken as line "B" and the half plane is the upper half ("y" > 0 ) of this plane. The hyperboloid model or Lorentz model employs a 2-dimensional hyperboloid of revolution (of two sheets, but using one) embedded in 3-dimensional Minkowski space. This model is generally credited to Poincaré, but Reynolds says that

neg: Geographic coordinate system Geographic coordinate system A geographic coordinate system is a coordinate system that enables every location on Earth to be specified by a set of numbers, letters or symbols. The coordinates are often chosen such that one of the numbers represents a vertical position and two or three of the numbers represent a horizontal position; alternatively, a geographic position may be expressed in a combined 3D Cartesian vector. A common choice of coordinates is latitude, longitude and elevation. To specify a location on a plane requires a map projection. The invention of a geographic coordinate system is generally credited to Eratosthenes

neg: Spherical basis e, e, e, and coordinates "A", "A", "A": or any other coordinate system with associated basis set of vectors. From this extend the scalars to allow multiplication by complex numbers, so that we are now working in formula_1 rather than formula_2. In the spherical bases denoted e, e, e, and associated coordinates with respect to this basis, denoted "A", "A", "A", the vector A is: where the spherical basis vectors can be defined in terms of the Cartesian basis using complex-valued coefficients in the "xy" plane: \mathbf{e}_{-} & = +\frac{1}{\sqrt{2}}\mathbf{e}_x - \frac{i}{\sqrt{2}}\mathbf{e}_y \\ in which "i" denotes the imaginary unit,

neg: Direct sum but is usually strictly smaller when the index set formula_23 is infinite, because direct products do not have the restriction that all but finitely many coordinates must be zero. For example, the "xy"-plane, a two-dimensional vector space, can be thought of as the direct sum of two one-dimensional vector spaces, namely the "x" and "y" axes. In this direct sum, the "x" and "y" axes intersect only at the origin (the zero vector). Addition is defined coordinate-wise, that is formula_24, which is the same as vector addition. Given two objects formula_4 and formula_5, their direct sum is written as formula_6.

neg: 2D computer graphics reversing the direction of the vector: Similarly, the product of translation matrices is given by adding the vectors: Because addition of vectors is commutative, multiplication of translation matrices is therefore also commutative (unlike multiplication of arbitrary matrices). In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. rotates points in the "xy"-Cartesian plane counterclockwise through an angle "θ" about the origin of the Cartesian coordinate system. To perform the rotation using a rotation matrix "R", the position of each point must be represented by a column vector v, containing the coordinates

neg: Etendue in figure "optical momentum". In a spherical coordinate system p may be written as from which and therefore, for an infinitesimal area d"S" = d"x" d"y" on the "xy" plane immersed in a medium of refractive index "n", the etendue is given by which is an infinitesimal volume in phase space "x", "y", "p", "q". Conservation of etendue in phase space is the equivalent in optics to Liouville's theorem in classical mechanics. Etendue as volume in phase space is commonly used in nonimaging optics. Consider an infinitesimal surface d"S", immersed in a medium of refractive index "n" crossed by (or

neg: Coordinate-measuring machine movement of the bridge or gantry along the granite table forms one axis of the XY plane. The bridge of the gantry contains a carriage which traverses between the inside and outside legs and forms the other X or Y horizontal axis. The third axis of movement (Z axis) is provided by the addition of a vertical quill or spindle which moves up and down through the center of the carriage. The touch probe forms the sensing device on the end of the quill. The movement of the X, Y and Z axes fully describes the measuring envelope. Optional rotary

neg: Linear function (calculus) "x" via a linear function: formula_60. In the "xy"-coordinate plane, the possible values of formula_61 form a line, the graph of the function formula_6. If formula_63 in the original equation, the resulting line formula_64 is vertical, and cannot be written as formula_32. The features of the graph formula_60 can be interpreted in terms of the variables "x" and "y". The "y"-intercept is the initial value formula_10 at formula_68. The slope "a" measures the rate of change of the output "y" per unit change in the input "x". In the graph, moving one unit to the right (increasing "x" by 1)

neg: Flight dynamics (fixed-wing aircraft) produced by the body lift. A good attempt for the induced drag coefficient is to assume a parabolic dependency of the lift Aerodynamic efficiency is now calculated as: If the configuration of the plane is symmetrical respect to the XY plane, minimum drag coefficient equals to the parasitic drag of the plane. In case the configuration is asymmetrical respect to the XY plane, however, minimum flag differs from the parasitic drag. On these cases, a new approximate parabolic drag equation can be traced leaving the minimum drag value at zero lift value. The Coefficient of pressure varies with Mach number

neg: Rotation of axes situated conveniently with respect to the axes, the coordinate system should be changed to place the curve at a convenient and familiar location and orientation. The process of making this change is called a transformation of coordinates. The solutions to many problems can be simplified by rotating the coordinate axes to obtain new axes through the same origin. The equations defining the transformation in two dimensions, which rotates the "xy" axes counterclockwise through an angle formula_1 into the "x'y<nowiki>'</nowiki>" axes, are derived as follows. In the "xy" system, let the point "P" have polar coordinates formula_4. Then, in the "x'y<nowiki>'</nowiki>"

neg: Klein bottle can start with a Möbius strip and curl it to bring the edge to the midline; since there is only one edge, it will meet itself there, passing through the midline. It has a particularly simple parametrization as a "figure-8" torus with a half-twist: for 0 ≤ "θ" < 2π, 0 ≤ "v" < 2π and "r" > 2. In this immersion, the self-intersection circle (where sin("v") is zero) is a geometric circle in the "xy" plane. The positive constant "r" is the radius of this circle. The parameter "θ" gives the angle in the "xy" plane as well as

neg: Coordinate systems for the hyperbolic plane Coordinate systems for the hyperbolic plane In the hyperbolic plane, as in the Euclidean plane, each point can be uniquely identified by two real numbers. Several qualitatively different ways of coordinatizing the plane in hyperbolic geometry are used. This article tries to give an overview of several coordinate systems in use for the two-dimensional hyperbolic plane. In the descriptions below the constant Gaussian curvature of the plane is −1. Sinh, cosh and tanh are hyperbolic functions. The polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference

neg: Laughlin wavefunction the approximation that all of the electrons lie in the LLL. If formula_6 is the single particle wavefunction of the LLL state with the lowest orbital angular momentum, then the Laughlin ansatz for the multiparticle wavefunction is where position is denoted by in (Gaussian units) and formula_10 and formula_11 are coordinates in the xy plane. Here formula_12 is Planck's constant, formula_13 is the electron charge, formula_14 is the total number of particles, and formula_15 is the magnetic field, which is perpendicular to the xy plane. The subscripts on z identify the particle. In order for the wavefunction to describe fermions,

neg: Analytic geometry is represented by its distance "r" from the origin and its angle "θ" from the polar axis. In cylindrical coordinates, every point of space is represented by its height "z", its radius "r" from the "z"-axis and the angle "θ" its projection on the "xy"-plane makes with respect to the horizontal axis. In spherical coordinates, every point in space is represented by its distance "ρ" from the origin, the angle "θ" its projection on the "xy"-plane makes with respect to the horizontal axis, and the angle "φ" that it makes with respect to the "z"-axis. The names of the angles

neg: Incremental sheet forming is generally implemented by clamping a sheet in the XY plane, which is free to move along the Z axis. The tool moves in the XY plane and is coordinated with movements in the Z axis to create the desired part. It is often convenient to retrofit a CNC milling machine to accommodate the process. Spherical, flat-bottomed, and parabolic tool profiles can be used to achieve differing surface finishes and forming limits. The machine employs a combination of stretch forming by drawing the sheet incrementally down over a die, with the CNC tool approach described above. This is said to

neg: State Plane Coordinate System state plane zone boundary. The Seattle metropolitan area is an example of this. King County, which includes the City of Seattle, uses the "Washington State Plane North" coordinate system, while Pierce County, which includes the City of Tacoma, uses "Washington State Plane South". Thus any regional agency that wants to combine regional data from local governments has to transform at least some data into a common coordinate system. State Plane Coordinate System The State Plane Coordinate System (SPS or SPCS) is a set of 124 geographic zones or coordinate systems designed for specific regions of the United States. Each state

neg: XY gonadal dysgenesis of female characteristics. The consequences of streak gonads to a person with Swyer syndrome: XY gonadal dysgenesis Swyer syndrome, or XY gonadal dysgenesis, is a type of hypogonadism in a person whose karyotype is 46,XY. The person is externally female with streak gonads, and if left untreated, will not experience puberty. Such gonads are typically surgically removed (as they have a significant risk of developing tumors) and a typical medical treatment would include hormone replacement therapy. The syndrome was named by Gerald Swyer, an endocrinologist, based in London. Types include: Seven other genes have been identified with probable associations that

neg: Pulsed electron paramagnetic resonance the electron spins interact with their surroundings and the magnetization in the xy-plane will decay and eventually return to alignment with the z-axis. This relaxation process is described by the spin-lattice relaxation time T, which is a characteristic time needed by the magnetization to return to the z-axis, and by the spin-spin relaxation time T, which describes the vanishing time of the magnetization in the xy-plane. The spin-lattice relaxation results from the urge of the system to return to thermal equilibrium after it has been perturbed by the B pulse. Return of the magnetization parallel to B is achieved through

neg: XY gonadal dysgenesis XY gonadal dysgenesis Swyer syndrome, or XY gonadal dysgenesis, is a type of hypogonadism in a person whose karyotype is 46,XY. The person is externally female with streak gonads, and if left untreated, will not experience puberty. Such gonads are typically surgically removed (as they have a significant risk of developing tumors) and a typical medical treatment would include hormone replacement therapy. The syndrome was named by Gerald Swyer, an endocrinologist, based in London. Types include: Seven other genes have been identified with probable associations that are as-yet less clearly understood. There are several forms of gonadal dysgenesis. The term

neg: State Plane Coordinate System feet". This information is needed in order to accurately transform data from one coordinate system to another. The main problem with the state plane coordinate system is that each zone uses a different coordinate system. This is not a major problem as long as one's needs are within the boundaries of a given state plane zone, as is the case with most county and city governments. However, the need to transform spatial data from one coordinate system to another can be burdensome. Sometimes a regional area of interest — such as a metropolitan area covering several counties — crosses a

neg: Charts on SO(3) the point is moved from the north to south pole, the latitude ranges from zero to 180 degrees, while the angle of rotation ranges from 0 to 360 degrees. (the "longitude" of a point then represents a particular axis of rotation.) Note however that this set of rotations is not closed under composition. Two successive rotations with axes in the "xy"-plane will not necessarily give a rotation whose axis lies in the "xy"-plane, and thus cannot be represented as a point on the sphere. This will not be the case with a general rotation in 3-space, which do form a

neg: Vector notation height). The first distance, usually represented as "r" or "ρ" (the Greek letter rho), is the magnitude of the projection of the vector onto the "xy"-plane. The angle, usually represented as "θ" or "φ" (the Greek letter phi), is measured as the offset from the line collinear with the "x"-axis in the positive direction; the angle is typically reduced to lie within the range formula_18. The second distance, usually represented as "h" or "z", is the distance from the "xy"-plane to the endpoint of the vector. Cylindrical vectors are specified like polar vectors, where the second distance component is concatenated

neg: Vector notation angle symbol (formula_20) is optional. A cylindrical vector can also be specified directly, using simplified autonomous equations that define "r" (or "ρ"), "θ" (or "φ"), and "h" (or "z"). Consistency should be used when choosing the names to use for the variables; "ρ" should not be mixed with "θ" and so on. A three-dimensional vector, the magnitude of whose projection onto the "xy"-plane is 5 units, whose angle from the positive "x"-axis is "π"/9 radians (20°), and whose height from the "xy"-plane is 3 units can be specified in any of the following forms: A spherical vector is another method

neg: XY sex-determination system Edmund Beecher Wilson are credited with independently discovering, in 1905, the chromosomal XY sex-determination system, i.e. the fact that males have XY sex chromosomes and females have XX sex chromosomes. The first clues to the existence of a factor that determines the development of testis in mammals came from experiments carried out by Alfred Jost, who castrated embryonic rabbits in utero and noticed that they all developed as female. In 1959, C. E. Ford and his team, in the wake of Jost's experiments, discovered that the Y chromosome was needed for a fetus to develop as male when they examined

neg: Pulsed electron paramagnetic resonance the magnetization vector rotates in the xy-plane, either faster or slower than the microwave magnetic field B. The rotation rate is governed by the frequency difference Δω. If Δω is 0 then the microwave field rotates as fast as the magnetization vector and both appear to be stationary to each other. If Δω>0 then the magnetization rotates faster than the microwave field component in a counter-clockwise motion and if Δω<0 then the magnetization is slower and rotates clockwise. This means that the individual frequency components of the EPR spectrum, will appear as magnetization components rotating in the xy-plane with the

neg: Vector notation as a third component to form ordered triplets (again, a subset of ordered set notation) and matrices. The angle may be prefixed with the angle symbol (formula_20); the distance-angle-distance combination distinguishes cylindrical vectors in this notation from spherical vectors in similar notation. A three-dimensional cylindrical vector "v" can be represented as any of the following, using either ordered triplet or matrix notation: Where "r" is the magnitude of the projection of v onto the "xy"-plane, "θ" is the angle between the positive "x"-axis and v, and "h" is the height from the "xy"-plane to the endpoint of "v". Again, the

neg: Charts on SO(3) we specify the point at the north pole to represent the identity rotation. In the case of the identity rotation, no axis of rotation is defined, and the angle of rotation (zero) is irrelevant. A rotation with its axis contained in the "xy"-plane and a very small rotation angle can be specified by a slice through the sphere parallel to the "xy"-plane and very near the north pole. The circle defined by this slice will be very small, corresponding to the small angle of the rotation. As the rotation angles become larger, the slice moves southward, and the circles become

neg: Quaternions and spatial rotation of a circle representing a particular rotation angle will be half of the angle represented by that rotation, since as the point is moved from the north to south pole, the latitude ranges from zero to 180 degrees, while the angle of rotation ranges from 0 to 360 degrees. (the "longitude" of a point then represents a particular axis of rotation.) Note however that this set of rotations is not closed under composition. Two successive rotations with axes in the "xy" plane will not necessarily give a rotation whose axis lies in the "xy" plane, and thus cannot be represented

neg: Euler angles both the first and third elemental rotations (e.g., "z-x-z", or "z-x’-z″"). This implies a different definition for the line of nodes in the geometrical construction. In the proper Euler angles case it was defined as the intersection between two homologous Cartesian planes (parallel when Euler angles are zero; e.g. "xy" and "XY"). In the second one, it is defined as the intersection of two non-homologous planes (perpendicular when Euler angles are zero; e.g. "xy" and "YZ"). The three elemental rotations may occur either about the axes of the original coordinate system, which remains motionless (extrinsic rotations), or about the axes

neg: Celestial coordinate system into two equal hemispheres along a great circle. Rectangular coordinates, in appropriate units, are simply the cartesian equivalent of the spherical coordinates, with the same fundamental () plane and primary (-axis) direction. Each coordinate system is named after its choice of fundamental plane. The following table lists the common coordinate systems in use by the astronomical community. The fundamental plane divides the celestial sphere into two equal hemispheres and defines the baseline for the latitudinal coordinates, similar to the equator in the geographic coordinate system. The poles are located at ±90° from the fundamental plane. The primary direction is the

neg: State Plane Coordinate System State Plane Coordinate System The State Plane Coordinate System (SPS or SPCS) is a set of 124 geographic zones or coordinate systems designed for specific regions of the United States. Each state contains one or more state plane zones, the boundaries of which usually follow county lines. There are 110 zones in the contiguous US, with 10 more in Alaska, 5 in Hawaii, and one for Puerto Rico and US Virgin Islands. The system is widely used for geographic data by state and local governments. Its popularity is due to at least two factors. First, it uses a simple Cartesian

neg: Indoor golf and Y directions. The ball passes through the first set of sensors and its XY coordinate is captured. The ball then passes another set of sensors and the second XY coordinates are captured again. The two co-ordinates are then compared to determine the vertical launch angle, ball path and speed. The system is very accurate very expensive to construct as you need hundreds of optical sensors and emitters to catch the coordinates of the ball in both sensor positions. The main disadvantage is that the ball must always pass through both sensors with a fast shot so that there is

neg: Rotation of axes Rotation of axes In mathematics, a rotation of axes in two dimensions is a mapping from an "xy"-Cartesian coordinate system to an "x'y<nowiki>'</nowiki>"-Cartesian coordinate system in which the origin is kept fixed and the "x<nowiki>'</nowiki>" and "y<nowiki>'</nowiki>" axes are obtained by rotating the "x" and "y" axes counterclockwise through an angle formula_1. A point "P" has coordinates ("x", "y") with respect to the original system and coordinates ("x<nowiki>'</nowiki>", "y<nowiki>'</nowiki>") with respect to the new system. In the new coordinate system, the point "P" will appear to have been rotated in the opposite direction, that is, clockwise through the angle formula_1.

neg: Mohr–Coulomb theory space. The expressions for formula_2 and formula_3 can be generalized to three dimensions by developing expressions for the normal stress and the resolved shear stress on a plane of arbitrary orientation with respect to the coordinate axes (basis vectors). If the unit normal to the plane of interest is where formula_24 are three orthonormal unit basis vectors, and if the principal stresses formula_25 are aligned with the basis vectors formula_26, then the expressions for formula_27 are The Mohr–Coulomb failure criterion can then be evaluated using the usual expression for the six planes of maximum shear stress. The Mohr–Coulomb yield surface

neg: Scrabble made in 1989 did not affect game play. The major changes in 1999 were as follows: In the notation system common in tournament play, columns are labeled with the letters "A-O" and rows with the numbers "1-15". (On "Scrabble" boards manufactured by Mattel as well as on the Internet Scrabble Club, rows are lettered while columns are numbered instead.) A play is usually identified in the format "xy WORD score" or "WORD xy score", where "x" denotes the column or row on which the play's main word extends, "y" denotes the second coordinate of the main word's first letter, and

neg: Ritz ballistic theory acceleration of the particles relative to each other (formula_5). This gives us an equation of the form: where the coefficients formula_7, formula_8 and formula_9 are independent of the coordinate system and are functions of formula_10 and formula_11. The stationary coordinates of the observer relate to the moving frame of the charge as follows Developing the terms in the force equation, we find that the density of particles is given by The tangent plane of the shell of emitted particles in the stationary coordinate is given by the Jacobian of the transformation from formula_14 to formula_15: We can also develop expressions

neg: Galactic coordinate system Galactic coordinate system The galactic coordinate system is a celestial coordinate system in spherical coordinates, with the Sun as its center, the primary direction aligned with the approximate center of the Milky Way galaxy, and the fundamental plane parallel to an approximation of the galactic plane but offset to its north. It uses the right-handed convention, meaning that coordinates are positive toward the north and toward the east in the fundamental plane. Longitude (symbol ) measures the angular distance of an object eastward along the galactic equator from the galactic center. Analogous to terrestrial longitude, galactic longitude is usually measured

neg: Translation of axes Translation of axes In mathematics, a translation of axes in two dimensions is a mapping from an "xy"-Cartesian coordinate system to an "x'y<nowiki>'</nowiki>"-Cartesian coordinate system in which the "x<nowiki>'</nowiki>" axis is parallel to the "x" axis and "k" units away, and the "y<nowiki>'</nowiki>" axis is parallel to the "y" axis and "h" units away. This means that the origin "O<nowiki>'</nowiki>" of the new coordinate system has coordinates ("h", "k") in the original system. The positive "x<nowiki>'</nowiki>" and "y<nowiki>'</nowiki>" directions are taken to be the same as the positive "x" and "y" directions. A point "P" has coordinates ("x", "y") with

neg: Resolution of singularities the previous blowups used to produce it. Some resolution methods (in characteristic 0) are functorial for all smooth morphisms. However it is not possible to find a strong resolution functorial for all (possibly non-smooth) morphisms. An example is given by the map from the affine plane "A" to the conical singularity "x" + "y" = "z" taking ("X","Y") to (2"XY", "X" − "Y", "X" + "Y"). The "XY"-plane is already nonsingular so should not be changed by resolution, and any resolution of the conical singularity factorizes through the minimal resolution given by blowing up the singular point. However the rational

neg: Azimuth installation, azimuth is one of the two coordinates. The other is altitude, sometimes called elevation above the horizon. See also: Sat finder. In mathematics, the azimuth angle of a point in cylindrical coordinates or spherical coordinates is the anticlockwise angle between the positive "x"-axis and the projection of the vector onto the "xy"-plane. The angle is the same as an angle in polar coordinates of the component of the vector in the "xy"-plane and is normally measured in radians rather than degrees. As well as measuring the angle differently, in mathematical applications theta, "θ", is very often used to represent

neg: Normal mapping –1} would be remapped to the {128, 128, 255} values, giving that kind of sky blue color seen in normal maps (blue (z) coordinate is perspective (deepness) coordinate and RG-xy flat coordinates on screen). {0.3, 0.4, –0.866} would be remapped to the ({0.3, 0.4, –0.866}/2+{0.5, 0.5, 0.5})*255={0.15+0.5, 0.2+0.5, -0.433+0.5}*255={0.65, 0.7, 0.067}*255={166, 179, 17} values (formula_1). The sign of the z-coordinate (blue channel) must be flipped to match the normal map's normal vector with that of the eye (the viewpoint or camera) or the light vector. Since negative z values mean that the vertex is in front of the camera (rather

neg: Schlieren imaging traveling parallel to the Z-axis, the XY planes are iso-phase manifolds (regions of constant phase; the phase does not depend on coordinates (x,y)). However, when the wave emerges from the acoustic field, XY planes are not iso-phase manifoldes anymore; the information about the accumulated pressure along each (x,y) line resides in the phase of the emerging radiation, forming a phase image (phasor) in the XY plane. The phase information is given by the Raman-Nath parameter: with formula_2 - the piezooptic coefficient, formula_3 the optical wavelength and formula_4 the three-dimensional pressure field. The schlieren technique converts the phase information into an

neg: Celestial coordinate system of the Sun). The system is primarily used for computing the positions of planets and other solar system bodies, as well as defining their orbital elements. The galactic coordinate system uses the approximate plane of our galaxy as its fundamental plane. The solar system is still the center of the coordinate system, and the zero point is defined as the direction towards the galactic center. Galactic latitude resembles the elevation above the galactic plane and galactic longitude determines direction relative to the center of the galaxy. The supergalactic coordinate system corresponds to a fundamental plane that contains a higher than

neg: Rockworks isosurface within a block model can. Stated differently, when dealing with grids, there can only be one z-value for any given xy coordinate. On the other hand, when dealing with block models, there can only be one g-value for any given xyz coordinate. Another major difference is that gridding is computationally fast while block modeling can be very slow. Consider the evaluation of a clay deposit in which the only important parameter is the thickness of the clay (i.e. the clay grade is homogeneous or “anisotropic”). Variations in the clay thickness encountered within nine boreholes are depicted by Figure 1.

neg: Mile American Datum of 1983 (NAD83), which replaced the NAD27, is defined in meters. State Plane Coordinate Systems were then updated, but the National Geodetic Survey left individual states to decide which (if any) definition of the foot they would use. All State Plane Coordinate Systems are defined in meters, and 42 of the 50 states only use the metre-based State Plane Coordinate Systems. However, eight states also have State Plane Coordinate Systems defined in feet, seven of them in U.S. Survey feet and one in international feet. State legislation in the U.S. is important for determining which conversion factor from

neg: Cartesian coordinate system plane with the thumb pointing up, the fingers point from the "x"-axis to the "y"-axis, in a positively oriented coordinate system. The other way of orienting the plane is following the "left hand rule", placing the left hand on the plane with the thumb pointing up. When pointing the thumb away from the origin along an axis towards positive, the curvature of the fingers indicates a positive rotation along that axis. Regardless of the rule used to orient the plane, rotating the coordinate system will preserve the orientation. Switching any two axes will reverse the orientation, but switching both will

neg: Abbe sine condition At this point another coordinate transformation can be proposed ("i"."e"., the Abbe sine condition) relating the object plane wavenumber spectrum to the image plane wavenumber spectrum as to obtain the final equation for the image plane field in terms of image plane coordinates and image plane wavenumbers as: From Fourier optics, it is known that the wavenumbers can be expressed in terms of the spherical coordinate system as If a spectral component is considered for which formula_12, then the coordinate transformation between object and image plane wavenumbers takes the form This is another way of writing the Abbe sine condition,

neg: Polarization (waves) instead, with the zero azimuth (or position angle, as it is more commonly called in astronomy to avoid confusion with the horizontal coordinate system) corresponding to due north. Another coordinate system frequently used relates to the "plane of incidence". This is the plane made by the incoming propagation direction and the vector perpendicular to the plane of an interface, in other words, the plane in which the ray travels before and after reflection or refraction. The component of the electric field parallel to this plane is termed "p-like" (parallel) and the component perpendicular to this plane is termed "s-like" (from

neg: Cleavage (geology) are likely to rotate so that their mean orientation is in the same direction as the XY plane of finite strain. Mineral grains may fold if oriented perpendicular to shortening direction. Cleavage foliations may result due to stress-induced solution transfer by the redistribution of inequant mineral grains by pressure solution and recrystallization. This would also help to increase rotation of elongate and tabular mineral grains. Mica grains undergoing solution transfer will align in a preferred orientation. If the minerals grains affected by pressure solution are deformed through plastic crystal processes, the grain will be extended along the XY-plane of finite

neg: Translation of axes vertices (or vertex), and eccentricity. Solution: To complete the square in "x" and "y", write the equation in the form Complete the squares and obtain Define That is, the translation in equations () is made with formula_8 The equation in the new coordinate system is Divide equation () by 225 to obtain which is recognizable as an ellipse with formula_10 In the "x'y<nowiki>'</nowiki>"-system, we have: center formula_11; vertices formula_12; foci formula_13 In the "xy"-system, use the relations formula_14 to obtain: center formula_15; vertices formula_16; foci formula_17; eccentricity formula_18 For an "xyz"-Cartesian coordinate system in three dimensions, suppose that a second

neg: Supergalactic coordinate system the galactic system, and the middle column is the cross product (to complete the right handed coordinate system). Supergalactic coordinate system Supergalactic coordinates are coordinates in a spherical coordinate system which was designed to have its equator aligned with the supergalactic plane, a major structure in the local universe formed by the preferential distribution of nearby galaxy clusters (such as the Virgo cluster, the Great Attractor and the Pisces-Perseus supercluster) towards a (two-dimensional) plane. The supergalactic plane was recognized by Gérard de Vaucouleurs in 1953 from the Shapley-Ames Catalog, although a flattened distribution of nebulae had been noted by William

neg: Poincaré half-plane model Poincaré half-plane model In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H formula_1 , together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry. Equivalently the Poincaré half-plane model is sometimes described as a complex plane where the imaginary part (the "y" coordinate mentioned above) is positive. The Poincaré half-plane model is named after Henri Poincaré, but it originated with Eugenio Beltrami, who used it, along with the Klein model and the Poincaré disk model (due to Bernhard Riemann), to show that hyperbolic geometry was equiconsistent with Euclidean

neg: Cross section (geometry) joining opposite vertices, the cross-section can be either a point, a triangle or a hexagon. A related concept is that of a plane section, which is the curve of intersection of a plane with a "surface". Thus, a plane section is the boundary of a cross-section of a solid in a cutting plane. If a surface in a three-dimensional space is defined by a function of two variables, i.e., , the plane sections by cutting planes that are parallel to a coordinate plane (a plane determined by two coordinate axes) are called level curves or isolines. More specifically, cutting planes

neg: Lincoln Beachey flew over the lower falls of Niagara Falls, then above American Falls, before an estimated 150,000 spectators. While gradually climbing, Beachey circled his plane over the falls several times. After he completed this performance he dove down into the mists of the falls, within 6 meters (20 feet) of the surface of the Niagara River. Then he flew his plane under the Honeymoon Bridge 6 meters (20 feet) above the rapids and down the length of the Niagara Gorge. Local papers described his plane as looking like "a beat-up orange crate". Although Wilfred Parke is credited with developing "Parke's technique"

neg: State Plane Coordinate System Surveyor Magazine", January 2004 link). It has been revised several times since then. When computers began to be used for mapping and GIS, the state plane system's cartesian grid system and simplified calculations made spatial processing faster and spatial data easier to work with. Even though computer processing power has improved radically since the early days of GIS, the size of spatial datasets and the complexity of geoprocessing tasks being demanded of computers have also increased. Thus the state plane coordinate system is still useful. Originally, the state plane coordinate systems were based on the North American Datum of 1927

neg: Cylindrical coordinate system Cylindrical coordinate system A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis. The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point. The "origin" of the system is the point where all three coordinates can be given as zero. This is the intersection between the reference plane and the axis. The axis is

neg: Method of image charges principle, a conducting plane below multiple point charges can be replaced by the mirror images of each of the charges individually, with no other modifications necessary. The image of an electric dipole moment p at formula_1 above an infinite grounded conducting plane in the "xy"-plane is a dipole moment at formula_3 with equal magnitude and direction rotated azimuthally by π. That is, a dipole moment with Cartesian components formula_10 will have in image dipole moment formula_11. The dipole experiences a force in the "z" direction, given by and a torque in the plane perpendicular to the dipole and the conducting

neg: Cylindrical coordinate system variously called the "cylindrical" or "longitudinal" axis, to differentiate it from the "polar axis", which is the ray that lies in the reference plane, starting at the origin and pointing in the reference direction. The distance from the axis may be called the "radial distance" or "radius", while the angular coordinate is sometimes referred to as the "angular position" or as the "azimuth".The radius and the azimuth are together called the "polar coordinates", as they correspond to a two-dimensional polar coordinate system in the plane through the point, parallel to the reference plane. The third coordinate may be called the

neg: Cylindrical coordinate system a surface of constant radius (a vertical cylinder) is The surface element in a surface of constant azimuth (a vertical half-plane) is The surface element in a surface of constant height (a horizontal plane) is The del operator in this system leads to the following expressions for gradient, divergence, curl and Laplacian: The solutions to the Laplace equation in a system with cylindrical symmetry are called cylindrical harmonics. Cylindrical coordinate system A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a

neg: Villarceau circles that he said a Colonel Schoelcher had presented at a congress in 1891. For example, suppose the major radius of the torus is 5 and the minor radius is 3. That means that the torus is the union of certain circles of radius three whose centers are on a circle of radius five in the "xy" plane. Points on this torus satisfy this equation: Slicing with the "z" = 0 plane produces two concentric circles, "x" + "y" = 2 and "x" + "y" = 8. Slicing with the "x" = 0 plane produces two side-by-side circles, ("y" − 5)

neg: Quaternions and spatial rotation be described by a rotation by some angle about some axis; for our purposes, we will use an axis "vector" to establish handedness for our angle. Consider the special case in which the axis of rotation lies in the "xy" plane. We can then specify the axis of one of these rotations by a point on a circle through which the vector crosses, and we can select the radius of the circle to denote the angle of rotation. Similarly, a rotation whose axis of rotation lies in the "xy" plane can be described as a point on a sphere of

neg: Curvilinear coordinates are curved. Well-known examples of curvilinear coordinate systems in three-dimensional Euclidean space (R) are Cartesian, cylindrical and spherical polar coordinates. A Cartesian coordinate surface in this space is a coordinate plane; for example "z" = 0 defines the "x"-"y" plane. In the same space, the coordinate surface "r" = 1 in spherical polar coordinates is the surface of a unit sphere, which is curved. The formalism of curvilinear coordinates provides a unified and general description of the standard coordinate systems. Curvilinear coordinates are often used to define the location or distribution of physical quantities which may be, for example, scalars,

neg: Coordinate systems for the hyperbolic plane to the point (0,0,1). The point P with axial coordinates ("x", "y") is mapped to Gyrovector space From Gyrovector space#triangle center The study of triangle centers traditionally is concerned with Euclidean geometry, but triangle centers can also be studied in hyperbolic geometry. Using gyrotrigonometry, expressions for trigonometric barycentric coordinates can be calculated that have the same form for both euclidean and hyperbolic geometry. In order for the expressions to coincide, the expressions must "not" encapsulate the specification of the anglesum being 180 degrees. Coordinate systems for the hyperbolic plane In the hyperbolic plane, as in the Euclidean plane, each point

neg: Airy function Stegun, 1954) and (Olver, 1974). We can extend the definition of the Airy function to the complex plane by where the integral is over a path "C" starting at the point at infinity with argument −π/3 and ending at the point at infinity with argument π/3. Alternatively, we can use the differential equation "y"′′ − "xy" = 0 to extend Ai("x") and Bi("x") to entire functions on the complex plane. The asymptotic formula for Ai("x") is still valid in the complex plane if the principal value of "x" is taken and "x" is bounded away from the negative real axis.

neg: Spherical coordinate system Spherical coordinate system In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the "radial distance" of that point from a fixed origin, its "polar angle" measured from a fixed zenith direction, and the "azimuth angle" of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system. The radial distance is also called the "radius" or

neg: Supergalactic coordinate system Supergalactic coordinate system Supergalactic coordinates are coordinates in a spherical coordinate system which was designed to have its equator aligned with the supergalactic plane, a major structure in the local universe formed by the preferential distribution of nearby galaxy clusters (such as the Virgo cluster, the Great Attractor and the Pisces-Perseus supercluster) towards a (two-dimensional) plane. The supergalactic plane was recognized by Gérard de Vaucouleurs in 1953 from the Shapley-Ames Catalog, although a flattened distribution of nebulae had been noted by William Herschel over 200 years earlier. Vera Rubin had also identified the supergalactic plane in the 1950s, but her

neg: Spherical coordinate system or , this reduces to vector calculus in polar coordinates. Spherical coordinate system In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the "radial distance" of that point from a fixed origin, its "polar angle" measured from a fixed zenith direction, and the "azimuth angle" of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate

neg: Graham scan Graham scan Graham's scan is a method of finding the convex hull of a finite set of points in the plane with time complexity O("n" log "n"). It is named after Ronald Graham, who published the original algorithm in 1972. The algorithm finds all vertices of the convex hull ordered along its boundary. It uses a stack to detect and remove concavities in the boundary efficiently. The first step in this algorithm is to find the point with the lowest y-coordinate. If the lowest y-coordinate exists in more than one point in the set, the point with the lowest x-coordinate

neg: Geographic coordinate system local tangent plane can be defined based on the vertical and horizontal dimensions. The vertical coordinate can point either up or down. There are two kinds of conventions for the frames: In many targeting and tracking applications the local ENU Cartesian coordinate system is far more intuitive and practical than ECEF or geodetic coordinates. The local ENU coordinates are formed from a plane tangent to the Earth's surface fixed to a specific location and hence it is sometimes known as a "Local Tangent" or "Local Geodetic" plane. By convention the east axis is labeled formula_13, the north formula_14 and the

neg: Radius axis, to differentiate it from the "polar axis", which is the ray that lies in the reference plane, starting at the origin and pointing in the reference direction. The distance from the axis may be called the "radial distance" or "radius", while the angular coordinate is sometimes referred to as the "angular position" or as the "azimuth". The radius and the azimuth are together called the "polar coordinates", as they correspond to a two-dimensional polar coordinate system in the plane through the point, parallel to the reference plane. The third coordinate may be called the "height" or "altitude" (if the

neg: Celestial coordinate system nutation. The fundamental plane is the plane of the Earth's orbit, called the ecliptic plane. There are two principal variants of the ecliptic coordinate system: geocentric ecliptic coordinates centered on the Earth and heliocentric ecliptic coordinates centered on the center of mass of the solar system. The geocentric ecliptic system was the principal coordinate system for ancient astronomy and is still useful for computing the apparent motions of the Sun, Moon, and planets. The heliocentric ecliptic system describes the planets' orbital movement around the Sun, and centers on the barycenter of the solar system (i.e. very close to the center

neg: Translation of axes of axes may be used to simplify second-degree equations, thereby making evident the nature of certain quadric surfaces. The principal tool in this process is "completing the square." Use a translation of coordinates to identify the quadric surface Solution: Write the equation in the form Complete the square to obtain Introduce the translation of coordinates The equation of the surface takes the form which is recognizable as the equation of an ellipsoid. Translation of axes In mathematics, a translation of axes in two dimensions is a mapping from an "xy"-Cartesian coordinate system to an "x'y<nowiki>'</nowiki>"-Cartesian coordinate system in which the

neg: Rotation of axes any finite number of dimensions, a rotation matrix formula_28 is an orthogonal matrix that differs from the identity matrix in at most four elements. These four elements are of the form for some formula_1 and some "i" ≠ "j". Find the coordinates of the point formula_32 after the positive "w" axis has been rotated through the angle formula_33, or 15°, into the positive "z" axis. Solution: Rotation of axes In mathematics, a rotation of axes in two dimensions is a mapping from an "xy"-Cartesian coordinate system to an "x'y<nowiki>'</nowiki>"-Cartesian coordinate system in which the origin is kept fixed and the

neg: Euler angles by defining the line of nodes as the intersection of the planes "xy" and "XY" (it can also be defined as the common perpendicular to the axes "z" and "Z" and then written as the vector product "N" = "z" formula_1 "Z"). Using it, the three Euler angles can be defined as follows: Euler angles between two reference frames are defined only if both frames have the same handedness. Intrinsic rotations are elemental rotations that occur about the axes of a coordinate system "XYZ" attached to a moving body. Therefore, they change their orientation after each elemental rotation. The "XYZ"

neg: Resolution of singularities map from the "XY"-plane to this blowup does not extend to a regular map. Minimal resolutions (resolutions such that every resolution factors through them) exist in dimensions 1 and 2, but not always in higher dimensions. The Atiyah flop gives an example in 3 dimensions of a singularity with no minimal resolution. Let "Y" be the zeros of "xy" = "zw" in A, and let "V" be the blowup of "Y" at the origin. The exceptional locus of this blowup is isomorphic to P×P, and can be blown down to P in 2 different ways, giving two small resolutions "X"

neg: Schwarzschild geodesics begins moving in that plane will remain in that plane indefinitely (the plane is totally geodesic). Therefore, we orient the coordinate system so that the orbit of the particle lies in that plane, and fix the "θ" coordinate to be π/2 so that the metric (of this plane) simplifies to Two constants of motion can then be identified (cf. the derivation given below). One is the total energy "E": and the other is the specific angular momentum: where L is the total angular momentum of the two bodies, and formula_6 is the reduced mass. When M»m, the reduced mass is

neg: Cauchy stress tensor continuum associated with a plane with normal unit vector n can be expressed as a function of the stress vectors on the planes perpendicular to the coordinate axes, "i.e." in terms of the components "σ" of the stress tensor σ. To prove this expression, consider a tetrahedron with three faces oriented in the coordinate planes, and with an infinitesimal area d"A" oriented in an arbitrary direction specified by a normal unit vector n (Figure 2.2). The tetrahedron is formed by slicing the infinitesimal element along an arbitrary plane n. The stress vector on this plane is denoted by T. The

neg: Coordinate system by converting the pair of cylindrical coordinates ("r", "z") to polar coordinates ("ρ", "φ") giving a triple ("ρ", "θ", "φ"). A point in the plane may be represented in "homogeneous coordinates" by a triple ("x", "y", "z") where "x"/"z" and "y"/"z" are the Cartesian coordinates of the point. This introduces an "extra" coordinate since only two are needed to specify a point on the plane, but this system is useful in that it represents any point on the projective plane without the use of infinity. In general, a homogeneous coordinate system is one where only the ratios of the coordinates

neg: Dimensional analysis orientation as the identity element, independent of the "symmetry of the problem". Physical quantities that are vectors have the orientation expected: a force or a velocity in the z-direction has the orientation of . For angles, consider an angle that lies in the z-plane. Form a right triangle in the z-plane with being one of the acute angles. The side of the right triangle adjacent to the angle then has an orientation and the side opposite has an orientation . Then, since we conclude that an angle in the xy-plane must have an orientation , which is not unreasonable. Analogous

neg: Cartesian coordinate system Leibniz. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Many other coordinate systems have been developed since Descartes, such as the polar coordinates for the plane, and the spherical and cylindrical coordinates for three-dimensional space. Choosing a Cartesian coordinate system for a one-dimensional space—that is, for a straight line—involves choosing a point "O" of the line (the origin), a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by "O" is the positive, and which is negative; we then say that the line "is

neg: Zoom H2n Handy Recorder reference speaker for listening the recordings. It can record in WAV format up to 24-bit/96 kHz and MP3 up to 320kbit/s. It has additional functions like Lo-cut Filter, Compressor/Limiter, Auto Gain, Pre-Rec, Auto-Rec, Tuner, Metronome, Variable Speed Playback built into it. The device can record in MS (mid-side), XY, MS+XY (2ch) and MS+XY (4ch) modes. If the 2.00 firmware update has been applied, it can also record in the 360-degree ambisonic format, along a horizontal plane. A recent 3.00 firmware update has also added support for 4ch input multitrack mode when connected as a USB audio interface, and the function

neg: Ecliptic coordinate system fundamental plane on the ecliptic plane, and the -axis toward the vernal equinox. The coordinates have a right-handed convention, that is, if one extends their right thumb upward, it simulates the -axis, their extended index finger the -axis, and the curl of the other fingers points generally in the direction of the -axis. These rectangular coordinates are related to the corresponding spherical coordinates by where is the obliquity of the ecliptic. Ecliptic coordinate system The ecliptic coordinate system is a celestial coordinate system commonly used for representing the apparent positions and orbits of Solar System objects. Because most planets (except