# Titlu referat: Coordonate carteziene

### Nivel referat: liceu

#### Descriere referat: Coordonate cartezieneA coordinate system is a method by which a setof numbers is used to locate the position of a point. The numbers are calledthe point's coordinates. In a coordinate system, a single point corresponds toeach set of coordinates. Coordinate systems are used in analytic geometry tostudy properties of geometric objects with algebraic techniques. When an object having a finite number ofdegrees of freedom is considered among all the objects of that kind, the objectin question can be conveniently characterized and distinguished from the otherobjects by a set of coordinatesÑthat is, a set of numbers, one for each degreeof freedom. For example, a point in a plane has two degrees of freedom, so thatthe point has two coordinates with respect to any coordinate system of theplane. There are many different coordinate systems.Usually the geometry and symmetry of a problem will suggest an appropriatecoordinate system. Common coordinate systems are Cartesian (after RenŽDescartes) coordinates and polar coordinates in two -dimensional space andCartesian, spherical, and cylindrical coordinates in three-dimensionalspace. Coordinate Systems in TwoDimensions Through an arbitrary point O in the plane, twomutually perpendicular lines, usually horizontal and vertical, are drawn. Thex-axis is taken to be horizontal, the y-axis is vertical, and point O is calledthe origin. The portion of the x-axis to the right of the origin is thepositive x-axis, and the part of the y-axis above the origin is called thepositive y-axis. The two axes (coordinate axes) divide the plane into fourquadrants: the upper right (first), the upper left (second), the lower left(third), and the lower right (fourth). The x-coordinate, or abscissa, of apoint P in the plane is theperpendicular distance of Pfrom the y-axis. It is positive if P is to the right of the y-axis, negative if P is to the left, and zero if P is on the y-axis. The y-coordinate, orordinate, of P is analogouslythe perpendicular distance of P from the x- axis. It is, respectively, positive, negative, or zeroif P is above, below, or onthe x-axis. The ordered pair (x, y) represents the coordinates of P in the coordinate system thus defined.The point P with coordinates(x, y) is symbolicallyrepresented as P (x, y). Thissystem is called a two- dimensional, or plane, Cartesian coordinatesystem. A polar coordinate system in two dimensions isa system determined by a fixed point O, called the pole, and an axis throughit, called the initial line. A point P in the plane can then be fixed by specifying two quantities: (1)the angle < through whichthe axis must be rotated in the counterclockwise direction so as to passthrough P, and (2) thepositive distance r of thepoint P from the pole. Thenotation P (r, <) is used torepresent P in polarcoordinates r and<. If P (x,y) is the Cartesian representation of P, and P (r,<) is the polarcoordinate representation of the same point, and if the origin and x-axis ofthe Cartesian system coincide, respectively, with the pole and the initial lineof the polar coordinate system, then the two systems are related by x = r cos<, y = r sin <, and r = the square root ofx6 + y6, tan < = y/x. These are the equations oftransformation from one system to another. Coordinate Systems in ThreeDimensions Three mutually perpendicular lines (thecoordinate axes) are drawn through an arbitrary point O, the origin, in space.The axes are called the x-axis, y-axis, and z-axis. The plane containing thex-axis and the y-axis is the xy-plane (a coordinate plane) and the z-axis is anormal (a line that is perpendicular) to this plane. The other two coordinateplanes are defined likewise. The x-coordinate of a point P is the perpendicular distance ofP from the yz-plane. They-coordinate and the z-coordinate are defined similarly. The three coordinateplanes divide all space into octants. If P is a point in the first octant, all the coordinates ofP are positive. The system thus described is athree-dimensional Cartesian system, and P (x, y,z) is the Cartesian representation of P with coordinates x, y, z with respectto a fixed frame of reference. For every point there corresponds uniquely a setof three real numbers, and vice versa. The spherical coordinate system in space is asystem that locates a point Pby its distance from a fixed point O (the pole), and by two angles thatdescribe the orientation of the segment OP. The coordinate system is fixed bytwo perpendicular half-lines through O. One of these is the polar axis. Theplane that contains the two half-lines is called the initial meridian plane.The spherical coordinates of P are (r, <,ñ), where r is the lengthof OP, < is the angle from the initialmeridian plane to the plane through the polar axis and OP, and ñ is the angle from the polar axis toOP. The spherical system isusually aligned with a Cartesian system in which the pole is the origin. Thepolar axis coincides with the z-axis, and the initial meridian plane with thexz-plane. The equations x = r sin ñ cos <, y = rsin <, and z = r cosñ express the relationbetween the two systems. A coordinate system consisting of a plane withpolar coordinates and a z-axis through the pole, or origin, perpendicular tothe plane is called a cylindrical polar coordinate system. A point P is located in space by its distancez from the plane and thepolar coordinates (r, <) of the foot of the perpendicularfrom P to the plane. The twosystems are related by the equations x = r cos <, y = sin <, and z = z. V. K. BalakrishnanBibliography: Fuller, Gordon,Analytic Geometry (1993);Leithold, Louis, Before Calculus, 2d ed. (1990). iiCopyright (c) 1997 Grolier Interactive Inc.

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