A cylindrical coordinate system is a threedimensional coordinate system, where each point is specified by the two polar coordinates of its perpendicular projection onto some fixed plane, and by its (signed) distance from that plane.
The polar coordinates may be called the radial distance or radius, and the angular position or azimuth, respectively. The third coordinate may be called the height or altitude (if the reference plane is considered horizontal), longitudinal position,^{[1]} or axial position.^{[2]} The line perpendicular to the reference plane that goes through its origin may be called the cylindrical axis or longitudinal axis.
Cylindrical coordinates are useful in connection with objects and phenomena that have some rotational symmetry about the longitudinal axis, such as water flow in a straight pipe with round crosssection, heat distribution in a metal cylinder, and so on.
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The three coordinates (ρ, φ, z) of a point P are defined as:
As in polar coordinates, the same point with cylindrical coordinates (ρ, φ, z) has infinitely many equivalent coordinates, namely (ρ, φ ± n×360°, z) and (−ρ, φ ± (2n + 1)×180°, z), where n is any integer. Moreover, if the radius ρ is zero, the azimuth is arbitrary.
In situations where one needs a unique set of coordinates for each point, one may restrict the radius to be nonnegative (ρ ≥ 0) and the azimuth φ to lie in a specific interval spanning 360°, such as (−180°,+180°] or [0,360°).
The notation for cylindrical coordinates is not uniform. The ISO standard 3111 recommends (ρ, φ, z), where ρ is the radial coordinate, φ the azimuth, and z the height. However, the radius is also often denoted r, the azimuth by θ or t, and the third coordinate by h or (if the cylindrical axis is considered horizontal) x, or any contextspecific letter.
In concrete situations, and in many mathematical illustrations, a positive angular coordinate is measured counterclockwise as seen from any point with positive height.
The cylindrical coordinate system is one of many threedimensional coordinate systems. The following formulae may be used to convert between them.
For the conversion between cylindrical and Cartesian coordinate systems, it is convenient to assume that the reference plane of the former is the Cartesian x–y plane (with equation z = 0) , and the cylindrical axis is the Cartesian z axis. Then the z coordinate is the same in both systems, and the correspondence between cylindrical (ρ,φ) and Cartesian (x,y) are the same as for polar coordinates, namely
in one direction, and
in the other. The arcsin function is the inverse of the sine function, and is assumed to return an angle in the range [−π/2,+π/2] = [−90°,+90°]. These formulas yield an azimuth φ in the range [−90°,+270°). For other formulas, see the polar coordinate article.
Many modern programming languages provide a function that will
compute the correct azimuth φ, in the range (−π, π], given
x and y, without the need to perform a case
analysis as above. For example, this function is called by atan2
(y,x) in the
C programming language, and atan
(y,x) in Common Lisp.
Spherical coordinates (radius r, elevation or inclination θ, azimuth φ), may be converted into cylindrical coordinates by:
θ is elevation:  θ is inclination:  
Cylindrical coordinates may be converted into spherical coordinates by:
θ is elevation:  θ is inclination:  
In many problems involving cylindrical polar coordinates, it is useful to know the line and volume elements; these are used in integration to solve problems involving paths and volumes.
The line element is
The volume element is
The surface element is
The del operator in this system is written as
and the Laplace operator Δ is defined by
The solutions to the Laplace equation in a system with cylindrical symmetry are called cylindrical harmonics.
