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p1033 Itf106tfs18 A coordinate system is a method by hich a set of numbers is used to locate the position of a point. The numbers are called the points coordinates. In a coordinate system, a single point corresponds to each set of coordinates. Coordinate systems are used in analytic geometry to study properties of geometric objects ith algebraic techniques.tpar tpar hen an object having a finite number of degrees of freedom is considered among all the objects of that kind, the object in question can be conveniently characterized and distinguished from the other objects by a set of coordinatestd1that is, a set of numbers, one for each degree of freedom. For example, a point in a plane has to degrees of freedom, so that the point has to coordinates ith respect to any coordinate system of the plane.tpar tpar There are many different coordinate systems. Usually the geometry and symmetry of a problem ill suggest an appropriate coordinate system. Common coordinate systems are Cartesian after Rent8e Descartes coordinates and polar coordinates in to -dimensional space and Cartesian, spherical, and cylindrical coordinates in three-dimensional space.tpar tpar SItbtf106tfs18 Coordinate Systems in To Dimensionstpar tpar SItf106tfs18 Through an arbitrary point O in the plane, to mutually perpendicular lines, usually horizontal and vertical, are dran. The x-axis is taken to be horizontal, the y-axis is vertical, and point O is called the origin. The portion of the x-axis to the right of the origin is the positive x-axis, and the part of the y-axis above the origin is called the positive y-axis. The to axes coordinate axes divide the plane into four quadrants the upper right first, the upper left second, the loer left third, and the loer right fourth. The x-coordinate, or abscissa, of a point SItitf106tfs18 PSItf106tfs18 in the plane is the perpendicular distance of SItitf106tfs18 PSItf106tfs18 from the y-axis. It is positive if SItitf106tfs18 PSItf106tfs18 is to the right of the y-axis, negative if SItitf106tfs18 PSItf106tfs18 is to the left, and zero if SItitf106tfs18 PSItf106tfs18 is on the y-axis. The y-coordinate, or ordinate, of SItitf106tfs18 PSItf106tfs18 is analogously the perpendicular distance of SItitf106tfs18 PSItf106tfs18 from the x- axis. It is, respectively, positive, negative, or zero if SItitf106tfs18 PSItf106tfs18 is above, belo, or on the x-axis. The ordered pair SItitf106tfs18 x, ySItf106tfs18 represents the coordinates of SItitf106tfs18 PSItf106tfs18 in the coordinate system thus defined. The point SItitf106tfs18 PSItf106tfs18 ith coordinates SItitf106tfs18 x, ySItf106tfs18 is symbolically represented as SItitf106tfs18 P x, ySItf106tfs18 . This system is called a to- dimensional, or plane, Cartesian coordinate system.tpar tpar A polar coordinate system in to dimensions is a system determined by a fixed point O, called the pole, and an axis through it, called the initial line. A point SItitf106tfs18 PSItf106tfs18 in the plane can then be fixed by specifying to quantities 1 the angle SItf107tfs18 SItf106tfs18 through hich the axis must be rotated in the counterclockise direction so as to pass through SItitf106tfs18 PSItf106tfs18 , and 2 the positive distance SItitf106tfs18 rSItf106tfs18 of the point SItitf106tfs18 PSItf106tfs18 from the pole. The notation SItitf106tfs18 P r, SItitf107tfs18 SItitf106tfs18 SItf106tfs18 is used to represent SItitf106tfs18 PSItf106tfs18 in polar coordinates SItitf106tfs18 rSItf106tfs18 and SItf107tfs18 SItf106tfs18 .tpar tpar If SItitf106tfs18 P x, ySItf106tfs18 is the Cartesian representation of SItitf106tfs18 PSItf106tfs18 , and SItitf106tfs18 P r, SItitf107tfs18 SItf106tfs18 is the polar coordinate representation of the same point, and if the origin and x-axis of the Cartesian system coincide, respectively, ith the pole and the initial line of the polar coordinate system, then the to systems are related by x r cos SItf107tfs18 SItf106tfs18 , y r sin SItf107tfs18 SItf106tfs18 , and r the square root of xSItf107tfs18 6SItf106tfs18 ySItf107tfs18 6SItf106tfs18 , tan SItf107tfs18 SItf106tfs18 yx. These are the equations of transformation from one system to another.tpar tpar SItbtf106tfs18 Coordinate Systems in Three Dimensionstpar tpar SItf106tfs18 Three mutually perpendicular lines the coordinate axes are dran through an arbitrary point O, the origin, in space. The axes are called the x-axis, y-axis, and z-axis. The plane containing the x-axis and the y-axis is the xy-plane a coordinate plane and the z-axis is a normal a line that is perpendicular to this plane. The other to coordinate planes are defined likeise. The x-coordinate of a point SItitf106tfs18 PSItf106tfs18 is the perpendicular distance of SItitf106tfs18 PSItf106tfs18 from the yz-plane. The y-coordinate and the z-coordinate are defined similarly. The three coordinate planes divide all space into octants. If SItitf106tfs18 PSItf106tfs18 is a point in the first octant, all the coordinates of SItitf106tfs18 PSItf106tfs18 are positive.tpar tpar The system thus described is a three-dimensional Cartesian system, and SItitf106tfs18 P x, y, zSItf106tfs18 is the Cartesian representation of SItitf106tfs18 PSItf106tfs18 ith coordinates SItitf106tfs18 xSItf106tfs18 , SItitf106tfs18 ySItf106tfs18 , SItitf106tfs18 zSItf106tfs18 ith respect to a fixed frame of reference. For every point there corresponds uniquely a set of three real numbers, and vice versa.tpar tpar The spherical coordinate system in space is a system that locates a point SItitf106tfs18 PSItf106tfs18 by its distance from a fixed point O the pole, and by to angles that describe the orientation of the segment OP. The coordinate system is fixed by to perpendicular half-lines through O. One of these is the polar axis. The plane that contains the to half-lines is called the initial meridian plane. The spherical coordinates of SItitf106tfs18 PSItf106tfs18 are r, SItf107tfs18 SItf106tfs18 , SItf107tfs18 tf1SItf106tfs18 , here r is the length of SItitf106tfs18 OPSItf106tfs18 , SItf107tfs18 SItf106tfs18 is the angle from the initial meridian plane to the plane through the polar axis and SItitf106tfs18 OPSItf106tfs18 , and SItf107tfs18 tf1SItf106tfs18 is the angle from the polar axis to SItitf106tfs18 OPSItf106tfs18 . The spherical system is usually aligned ith a Cartesian system in hich the pole is the origin. The polar axis coincides ith the z-axis, and the initial meridian plane ith the xz-plane. The equations x r sin SItf107tfs18 tf1SItf106tfs18 cos SItf107tfs18 SItf106tfs18 , y r sin SItf107tfs18 SItf106tfs18 , and z r cos SItf107tfs18 tf1SItf106tfs18 express the relation beteen the to systems.tpar tpar Stpard tqc tli0tri0tnoidctlpartfaautotrin0tlin0titap0 Itf106tfs18 IttshppictItpictIttpicproptshplid1025ItspItsn shapeTypeSItsv 75SSItspItsn fFlipHSItsv 0SSItspItsn fFlipVSItsv 0SSItspItsn fLineSItsv 0SSItspItsn fLayou...
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