co·or·di·nate

coordinate [past participle of coordinare, from Latin co- + ordinare 'to arrange']

The use of a coordinate system [坐标系] allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry [解析几何].

• Mathematical analysis [数学分析] is the part of mathematics in which functions and their generalizations are studied by the method of limits [极限].
• Analytic means the same as analytical. (mainly American English)

In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold [流形] such as Euclidean space [欧几里得空间, 欧式空间]. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple [多元组] and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system.

The simplest example of a coordinate system is the identification of points on a line with real numbers using the number line [数轴]. In this system, an arbitrary point O (the origin) is chosen on a given line. The coordinate of a point P is defined as the signed distance from O to P, where the signed distance is the distance taken as positive or negative depending on which side of the line P lies. Each point is given a unique coordinate and each real number is the coordinate of a unique point.

The prototypical example of a coordinate system is the Cartesian coordinate system [笛卡尔坐标系]. In the plane, two perpendicular lines are chosen and the coordinates of a point are taken to be the signed distances to the lines. In three dimensions, three mutually orthogonal planes are chosen and the three coordinates of a point are the signed distances to each of the planes. This can be generalized to create n coordinates for any point in n-dimensional Euclidean space.

Another common coordinate system for the plane is the polar coordinate system. A point is chosen as the pole and a ray from this point is taken as the polar axis. For a given angle θ, there is a single line through the pole whose angle with the polar axis is θ (measured counterclockwise from the axis to the line). Then there is a unique point on this line whose signed distance from the origin is r for given number r. For a given pair of coordinates (r, θ) there is a single point, but any point is represented by many pairs of coordinates. For example, (r, θ), (r, θ+2π) and (−r, θ+π) are all polar coordinates for the same point. The pole is represented by (0, θ) for any value of θ.

There are two common methods for extending the polar coordinate system to three dimensions. In the cylindrical coordinate system, a z-coordinate with the same meaning as in Cartesian coordinates is added to the r and θ polar coordinates giving a triple (r, θ, z). Spherical coordinates take this a step further by converting the pair of cylindrical coordinates (r, z) to polar coordinates (ρ, φ) giving a triple (ρ, θ, φ).

Because there are often many different possible coordinate systems for describing geometrical figures, it is important to understand how they are related. Such relations are described by coordinate transformations which give formulas for the coordinates in one system in terms of the coordinates in another system. For example, in the plane, if Cartesian coordinates (x, y) and polar coordinates (r, θ) have the same origin, and the polar axis is the positive x axis, then the coordinate transformation from polar to Cartesian coordinates is given by x = r cosθ and y = r sinθ.

六级/考研单词: coordinate, geometry, translate, vice, mathematics, elementary, abstract, arbitrary, dimension, mutual, ray, axis, clockwise, cylinder, triple, sphere, farther, convert, seldom, sinister