Editing Cylindrical coordinate system (section)

==Definition==
The three coordinates ({{mvar|[[Rho (letter)|ρ]]}}, {{mvar|[[Phi|φ]]}}, {{mvar|z}}) of a point {{mvar|P}} are defined as:
* The ''radial distance'' {{mvar|ρ}} is the [[Euclidean distance]] from the {{mvar|z}}-axis to the point {{mvar|P}}.
* The ''azimuth'' {{mvar|φ}} is the angle between the reference direction on the chosen plane and the line from the origin to the projection of {{mvar|P}} on the plane.
* The ''axial coordinate'' or ''height'' {{mvar|z}} is the signed distance from the chosen plane to the point {{mvar|P}}.

===Unique cylindrical coordinates===
As in polar coordinates, the same point with cylindrical coordinates {{math|(''ρ'', ''φ'', ''z'')}} has infinitely many equivalent coordinates, namely {{math|(''ρ'', ''φ'' ± ''n''×360°, ''z'')}} and {{math|(−''ρ'', ''φ'' ± (2''n'' + 1)×180°, ''z''),}} where {{mvar|n}} is any integer. Moreover, if the radius {{mvar|ρ}} is zero, the azimuth is arbitrary.

In situations where someone wants a unique set of coordinates for each point, one may restrict the radius to be [[non-negative]] ({{math|''ρ'' ≥ 0}}) and the azimuth {{mvar|φ}} to lie in a specific [[interval (mathematics)|interval]] spanning 360°, such as {{math|[−180°,+180°]}} or {{math|[0,360°]}}.

===Conventions===
The notation for cylindrical coordinates is not uniform. The [[International Organization for Standardization|ISO]] standard [[ISO 31-11|31-11]] recommends {{math|(''ρ'', ''φ'', ''z'')}}, where {{mvar|ρ}} is the radial coordinate, {{mvar|φ}} the azimuth, and {{mvar|z}} the height. However, the radius is also often denoted {{mvar|r}} or {{mvar|s}}, the azimuth by {{mvar|θ}} or {{mvar|t}}, and the third coordinate by {{mvar|h}} or (if the cylindrical axis is considered horizontal) {{mvar|x}}, or any context-specific letter.

[[File:Cylindrical coordinate surfaces.png|thumb|right|The [[coordinate surfaces]] of the cylindrical coordinates {{math|(''ρ'', ''φ'', ''z'')}}. The red [[Cylinder (geometry)|cylinder]] shows the points with {{math|''ρ'' {{=}} 2}}, the blue [[plane (mathematics)|plane]] shows the points with {{math|''z'' {{=}} 1}}, and the yellow half-plane shows the points with {{math|''φ'' {{=}} −60°}}. The {{mvar|z}}-axis is vertical and the {{mvar|x}}-axis is highlighted in green.  The three surfaces intersect at the point {{mvar|P}} with those coordinates (shown as a black sphere); the [[Cartesian coordinate system|Cartesian coordinates]] of {{mvar|P}} are roughly (1.0, −1.732, 1.0).]]

[[File:Cylindrical coordinate surfaces.gif|thumb|Cylindrical coordinate surfaces. The three orthogonal components, {{mvar|ρ}} (green), {{mvar|φ}} (red), and {{mvar|z}} (blue), each increasing at a constant rate. The point is at the intersection between the three colored surfaces.]]

In concrete situations, and in many mathematical illustrations, a positive angular coordinate is measured [[clockwise|counterclockwise]] as seen from any point with positive height.