“polar coordinates”的概念、定义、翻译、参考文献-科学参考

polar coordinates

Physics

A system used in analytical geometry to locate a point P, with reference to two or three axes. The distance of P from the origin is r, and the angle between the x-axis and the radius vector OP is θ‎, thus in two-dimensional polar coordinates the coordinates of P are (r,θ‎). If the Cartesian coordinates of P are (x,y) then x=rcosθ‎ and y=rsinθ‎.

In three dimensions the point P may be regarded as lying on the surface of a cylinder, giving cylindrical polar coordinates, or on the surface of a sphere, giving spherical polar coordinates. In the former the coordinates of P would be (r,θ‎,z); in the latter they would be (r,θ,ϕ‎).

Polar coordinates.

Mathematics

Suppose that a point O in the plane is chosen as origin, and let Ox be a directed line through O, with a given unit of length. For any point P in the plane, let r = |OP| and, if P is not O, let θ‎ be the angle (in radians) that OP makes with Ox, the angle being given a positive sense anticlockwise from Ox. The angle θ‎ satisfies 0 ≤ θ‎ < 2π‎. Then (r,θ‎) are the polar coordinates of P.

The polar coordinates of P

Suppose that Cartesian coordinates are taken with the same origin and the same unit of length, with positive x-axis along the directed line Ox. Then the Cartesian coordinates (x,y) of a point P can be found from (r,θ‎), by x = rcosθ‎, y = rsinθ‎. Conversely, the polar coordinates can be found from the Cartesian coordinates by $r = \sqrt{x^{2} + y^{2}}$ , and θ‎ is such that

$cos θ = \frac{x}{\sqrt{x^{2} + y^{2}}}, sin θ = \frac{y}{\sqrt{x^{2} + y^{2}}} .$

(Note that θ‎ = tan⁻¹(y/x) but this is not, of itself, sufficient to uniquely determine θ‎.) In certain circumstances, authors may allow r to be negative, in which case the polar coordinates (r,θ‎) give the same point as (–r,θ‎ + π‎).

In mechanics, it is useful, when a point P has polar coordinates (r,θ‎), to define unit vectors e_r and e_θ‎ by

$e_{r} = i cos θ + j sin θ and e_{θ} = - i sin θ + j cos θ,$

where i and j are unit vectors in the directions of the positive x- and y-axes. Then e_r is a unit vector along OP in the direction of increasing r, and e_θ‎ is a unit vector perpendicular to this in the direction of increasing θ‎. These vectors satisfy e_r∙e_θ‎ =0 and e_r×e_θ‎ =k, where k = i×j. See radial and transverse components.

Statistics

A coordinate system in a plane in which the position of a point P is specified by the distance r=OP, where O is a fixed point (the pole), and the angle θ between OP and a fixed line through O (the initial line). The polar coordinates are given as (r, θ), where r≥0, and −180°<θ≤180° or 0°≤θ<360°. Referred to Cartesian coordinates Oxy, with Ox along the initial line, Cartesian and polar coordinates are related by $polar coordinates$

Electronics and Electrical Engineering

A technique for giving two-dimensional coordinates (r,ϕ) of a point. Radial distance r is the distance along the horizontal x-axis, and polar angle ϕ describes an anti-clockwise rotation. See also cylindrical polar coordinates; spherical polar coordinates.
Polar coordinates

单词	polar coordinates
释义	polar coordinates Physics A system used in analytical geometry to locate a point P, with reference to two or three axes. The distance of P from the origin is r, and the angle between the x-axis and the radius vector OP is θ‎, thus in two-dimensional polar coordinates the coordinates of P are (r,θ‎). If the Cartesian coordinates of P are (x,y) then x=rcosθ‎ and y=rsinθ‎. In three dimensions the point P may be regarded as lying on the surface of a cylinder, giving cylindrical polar coordinates, or on the surface of a sphere, giving spherical polar coordinates. In the former the coordinates of P would be (r,θ‎,z); in the latter they would be (r,θ,ϕ‎). Polar coordinates. Mathematics Suppose that a point O in the plane is chosen as origin, and let Ox be a directed line through O, with a given unit of length. For any point P in the plane, let r = \|OP\| and, if P is not O, let θ‎ be the angle (in radians) that OP makes with Ox, the angle being given a positive sense anticlockwise from Ox. The angle θ‎ satisfies 0 ≤ θ‎ < 2π‎. Then (r,θ‎) are the polar coordinates of P. The polar coordinates of P Suppose that Cartesian coordinates are taken with the same origin and the same unit of length, with positive x-axis along the directed line Ox. Then the Cartesian coordinates (x,y) of a point P can be found from (r,θ‎), by x = rcosθ‎, y = rsinθ‎. Conversely, the polar coordinates can be found from the Cartesian coordinates by $r = \sqrt{x^{2} + y^{2}}$ , and θ‎ is such that $cos θ = \frac{x}{\sqrt{x^{2} + y^{2}}}, sin θ = \frac{y}{\sqrt{x^{2} + y^{2}}} .$ (Note that θ‎ = tan⁻¹(y/x) but this is not, of itself, sufficient to uniquely determine θ‎.) In certain circumstances, authors may allow r to be negative, in which case the polar coordinates (r,θ‎) give the same point as (–r,θ‎ + π‎). In mechanics, it is useful, when a point P has polar coordinates (r,θ‎), to define unit vectors e_r and e_θ‎ by $e_{r} = i cos θ + j sin θ and e_{θ} = - i sin θ + j cos θ,$ where i and j are unit vectors in the directions of the positive x- and y-axes. Then e_r is a unit vector along OP in the direction of increasing r, and e_θ‎ is a unit vector perpendicular to this in the direction of increasing θ‎. These vectors satisfy e_r∙e_θ‎ =0 and e_r×e_θ‎ =k, where k = i×j. See radial and transverse components. Statistics A coordinate system in a plane in which the position of a point P is specified by the distance r=OP, where O is a fixed point (the pole), and the angle θ between OP and a fixed line through O (the initial line). The polar coordinates are given as (r, θ), where r≥0, and −180°<θ≤180° or 0°≤θ<360°. Referred to Cartesian coordinates Oxy, with Ox along the initial line, Cartesian and polar coordinates are related by $polar coordinates$ Electronics and Electrical Engineering A technique for giving two-dimensional coordinates (r,ϕ) of a point. Radial distance r is the distance along the horizontal x-axis, and polar angle ϕ describes an anti-clockwise rotation. See also cylindrical polar coordinates; spherical polar coordinates. Polar coordinates
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