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A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis (axis L in the image opposite), the direction from the axis relative to a chosen reference direction (axis A), and the distance from a chosen reference plane perpendicular to the axis (plane ...
This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): . The polar angle is denoted by [,]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question.
Vectors are defined in cylindrical coordinates by (ρ, φ, z), where ρ is the length of the vector projected onto the xy-plane, φ is the angle between the projection of the vector onto the xy-plane (i.e. ρ) and the positive x-axis (0 ≤ φ < 2π), z is the regular z-coordinate. (ρ, φ, z) is given in Cartesian coordinates by:
Let (x, y, z) be the standard Cartesian coordinates, and (ρ, θ, φ) the spherical coordinates, with θ the angle measured away from the +Z axis (as , see conventions in spherical coordinates). As φ has a range of 360° the same considerations as in polar (2 dimensional) coordinates apply whenever an arctangent of it is taken. θ has a range ...
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). [8] 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 ...
Curvilinear coordinate systems, such as cylindrical or spherical coordinates, are often used in physical and geometric problems. Associated with any coordinate system is a natural choice of coordinate basis for vectors based at each point of the space, and covariance and contravariance are particularly important for understanding how the ...
Coordinate surfaces of parabolic cylindrical coordinates. The red parabolic cylinder corresponds to σ=2, whereas the yellow parabolic cylinder corresponds to τ=1. The blue plane corresponds to z=2. These surfaces intersect at the point P (shown as a black sphere), which has Cartesian coordinates roughly (2, -1.5, 2).
In three dimensions, any set of three-dimensional coordinates and their corresponding basis vectors can be used to define the location of a point in space—whichever is the simplest for the task at hand may be used. Commonly, one uses the familiar Cartesian coordinate system, or sometimes spherical polar coordinates, or cylindrical coordinates: