Cartesian coordinates


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Cartesian coordinates

(kärtē`zhən) [for René DescartesDescartes, René
, Lat. Renatus Cartesius, 1596–1650, French philosopher, mathematician, and scientist, b. La Haye. Descartes' methodology was a major influence in the transition from medieval science and philosophy to the modern era.
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], system for representing the relative positions of points in a plane or in space. In a plane, the point P is specified by the pair of numbers (x,y) representing the distances of the point from two intersecting straight lines, referred to as the x-axis and the y-axis. The point of intersection of these axes, which are called the coordinate axes, is known as the origin. In rectangular coordinates, the type most often used, the axes are taken to be perpendicular, with the x-axis horizontal and the y-axis vertical, so that the x-coordinate, or abscissa, of P is measured along the horizontal perpendicular from P to the y-axis (i.e., parallel to the x-axis) and the y-coordinate, or ordinate, is measured along the vertical perpendicular from P to the x-axis (parallel to the y-axis). In oblique coordinates the axes are not perpendicular; the abscissa of P is measured along a parallel to the x-axis, and the ordinate is measured along a parallel to the y-axis, but neither of these parallels is perpendicular to the other coordinate axis as in rectangular coordinates. Similarly, a point in space may be specified by the triple of numbers (x,y,z) representing the distances from three planes determined by three intersecting straight lines not all in the same plane; i.e., the x-coordinate represents the distance from the yz-plane measured along a parallel to the x-axis, the y-coordinate represents the distance from the xz-plane measured along a parallel to the y-axis, and the z-coordinate represents the distance from the xy-plane measured along a parallel to the z-axis (the axes are usually taken to be mutually perpendicular). Analogous systems may be defined for describing points in abstract spaces of four or more dimensions. Many of the curves studied in classical geometry can be described as the set of points (x,y) that satisfy some equation f(x,y)=0. In this way certain questions in geometry can be transformed into questions about numbers and resolved by means of analytic geometryanalytic geometry,
branch of geometry in which points are represented with respect to a coordinate system, such as Cartesian coordinates, and in which the approach to geometric problems is primarily algebraic.
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Cartesian Coordinates

 

a rectilinear system of coordinates in a plane or in space (usually with identical scales on both axes). R. Descartes himself used only a system of coordinates in a plane (generally oblique) in the work Geometry (1637). Often the Cartesian coordinates are understood to mean the rectangular Cartesian coordinates, while the general Cartesian coordinates are called an affine system of coordinates.

cartesian coordinates

[kär′tē·zhən kō′ȯrd·nəts]
(mathematics)
The set of numbers which locate a point in space with respect to a collection of mutually perpendicular axes.

Cartesian coordinates

(mathematics, graphics)
(After Renee Descartes, French philosopher and mathematician) A pair of numbers, (x, y), defining the position of a point in a two-dimensional space by its perpendicular projection onto two axes which are at right angles to each other. x and y are also known as the abscissa and ordinate.

The idea can be generalised to any number of independent axes.

Compare polar coordinates.
References in periodicals archive ?
(2002) Cue inhibitory effects on manual response to a visual target: Are they based on Cartesian coordinates? In: Da Silva, J.A., Matsushima, E.H.
The 3D Cartesian coordinate system on [E.sup.3] corresponds to choose the set B = {[e.sub.1],[e.sub.2],[e.sub.3]} as basis of [E.sup.3].
We again resort to Eqs 7-9 for constant stress sectors, along with Eqs 26 and 27 to get the stresses in the Cartesian coordinates in region III as
- Analysis of the field data by computer includes checking for errors and calculating Cartesian coordinates for all trees mapped to date, and proceeds as follows:
But all of these axes are useless without the Cartesian coordinate system.
It consists of the number of nodes N, type REAL arrays X, Y, and Z of length N containing the Cartesian coordinates of the nodes, and a linked list requiring approximately 13N storage locations and which contains the adjacency list (ordered sequence of neighbors) for each node:
However, in the improved SE-MLFMA, not only the RPs but also the aggregation, translation, and disaggregation processes are all operated in the Cartesian coordinates. Therefore, the improved SE-MLFMA totally obviates the need for transforming the RPs between different coordinates.
In Section 3, we consider a particular quantum system for applying the Mielniks and Marquette's method and obtain a superintegrable potential separable in Cartesian coordinates. In Section 4, we briefly review the master function formalism and then in Section 5, we use this approach to obtain integrable systems and particular cases of the superintegrable systems that satisfy the oscillator-like (Heisenberg) algebra with higher order integrals of motion in terms of the master function and weight function.
The surfaces of constant u describe confocal paraboloids about the polar axis, z in cartesian coordinates, that open in the direction of negative z or [theta] = [pi] rad and have a focus at the origin; the surfaces of constant v analogously describe confocal paraboloids that open in the direction of positive z, or [theta] = 0, and have also a focus at the origin.
where [[xi].sub.i], [[eta].sub.i] and [x'.sub.i], [y'.sub.i] (i = 1,2,3,4) denote the intrinsic coordinates and the local Cartesian coordinates of the element nodes, respectively.
After applying the two synchronization mechanisms described above, a correct translation from cylindrical to Cartesian coordinates can be performed.
Similarly, the only non-zero components of the stress tensor in cartesian coordinates are given by [14]