Editing Spherical coordinate system (section)

== Definition ==
To define a spherical coordinate system, one must designate an ''origin'' point in space, ''{{mvar|O}}'', and two orthogonal directions: the ''zenith reference'' direction and the ''azimuth reference'' direction. These choices determine a reference plane that is typically defined as containing the point of origin and the ''x{{ndash}} and y{{ndash}}axes'', either of which may be designated as the ''azimuth reference'' direction. The reference plane is perpendicular (orthogonal) to the zenith direction, and typically is designated "horizontal" to the zenith direction's "vertical". The spherical coordinates of a point {{mvar|P}} then are defined as follows:

* The ''radius'' or ''radial distance'' is the [[Euclidean distance]] from the origin ''{{mvar|O}}'' to ''{{mvar|P}}''.
* The ''inclination'' (or ''polar angle'') is the signed angle from the zenith reference direction to the line segment {{mvar|OP}}. (''Elevation'' may be used as the polar angle instead of ''inclination''; see below.)
* The ''[[azimuth]]'' (or ''azimuthal angle'') is the signed angle measured from the ''azimuth reference'' direction to the orthogonal projection of the radial line segment {{mvar|OP}} on the reference plane.
The sign of the azimuth is determined by designating the rotation that is the ''positive'' sense of turning about the zenith. This choice is arbitrary, and is part of the coordinate system definition. 
(If the inclination is either zero or 180 degrees (= {{pi}} radians), the azimuth is arbitrary. If the radius is zero, both azimuth and inclination are arbitrary.)

The [[Horizontal coordinate system|''elevation'']] is the signed angle from the x-y reference plane to the radial line segment {{mvar|OP}}, where positive angles are designated as upward, towards the zenith reference. ''Elevation'' is 90 degrees (= {{sfrac|{{pi}}|2}} radians) ''minus inclination''. Thus, if the inclination is 60 degrees (= {{sfrac|{{pi}}|3}} radians), then the elevation is 30 degrees (= {{sfrac|{{pi}}|6}} radians).

In [[linear algebra]], the [[Euclidean vector|vector]] from the origin {{mvar|O}} to the point {{mvar|P}} is often called the ''[[position vector]]'' of ''P''.

=== Conventions ===
Several different conventions exist for representing spherical coordinates and prescribing the naming order of their symbols. The 3-tuple number set <math>(r,\theta,\varphi)</math> denotes radial distance, the polar angle{{mdash}}"inclination", or as the alternative, "elevation"{{mdash}}and the azimuthal angle. It is the common practice within the physics convention, as specified by [[International Organization for Standardization|ISO]] standard [[ISO 80000-2|80000-2:2019]], and earlier in [[ISO 31-11]] (1992). 

''As stated above, this article describes the ISO "physics convention"{{mdash}}unless otherwise noted.''

However, some authors (including mathematicians) use the symbol ''ρ'' (rho) for radius, or radial distance, ''φ'' for inclination (or elevation) and ''θ'' for azimuth{{mdash}}while others keep the use of ''r'' for the radius; all which "provides a logical extension of the usual polar coordinates notation".<ref name="http://mathworld.wolfram.com/SphericalCoordinates.html">{{cite web |url=http://mathworld.wolfram.com/SphericalCoordinates.html |title=Spherical Coordinates |author=[[Eric W. Weisstein]] |publisher=[[MathWorld]] |date=2005-10-26 |access-date=2010-01-15}}</ref> As to order, some authors list the azimuth ''before'' the inclination (or the elevation) angle. Some combinations of these choices result in a [[right-hand rule|left-handed]] coordinate system. The standard "physics convention" 3-tuple set <math>(r,\theta,\varphi)</math> conflicts with the usual notation for two-dimensional [[polar coordinate system|polar coordinates]] and three-dimensional [[cylindrical coordinate system|cylindrical coordinates]], where {{mvar|θ}} is often used for the azimuth.<ref name="http://mathworld.wolfram.com/SphericalCoordinates.html" />

Angles are typically measured in [[Degree (angle)|degrees]] (°) or in [[radian]]s (rad), where 360°&nbsp;=&nbsp;2{{pi}} rad. The use of degrees is most common in geography, astronomy, and engineering, where radians are commonly used in mathematics and theoretical physics. The unit for radial distance is usually determined by the context, as occurs in applications of the 'unit sphere', see [[#Applications|applications]].

When the system is used to designate physical three-space, it is customary to assign positive to azimuth angles measured in the ''counterclockwise'' sense from the reference direction on the reference plane{{mdash}}as seen from the "zenith" side of the plane. This convention is used in particular for geographical coordinates, where the "zenith" direction is [[north]] and the positive azimuth (longitude) angles are measured eastwards from some [[prime meridian]].

{| class="wikitable" style="text-align:center"
|+ Major conventions
|-
! coordinates set order !! corresponding local geographical directions <br /> {{math|(''Z'', ''X'', ''Y'')}} !! right/left-handed
|-
| {{math|(''r'', ''θ''<sub>inc</sub>, ''φ''<sub>az,right</sub>)}} || {{math|(''U'', ''S'', ''E'')}} || right
|-
| {{math|(''r'', ''φ''<sub>az,right</sub>, ''θ''<sub>el</sub>)}}|| {{math|(''U'', ''E'', ''N'')}} || right
|-
| {{math|(''r'', ''θ''<sub>el</sub>, ''φ''<sub>az,right</sub>)}}|| {{math|(''U'', ''N'', ''E'')}} || left
|}
'''Note:''' [[Easting and northing|Easting ({{mvar|E}}), Northing ({{mvar|N}})]], Upwardness ({{mvar|U}}). In the case of {{math|(''U'', ''S'', ''E'')}} the local [[azimuth]] angle would be measured [[counterclockwise]] from {{mvar|S}} to {{mvar|E}}.

=== Unique coordinates ===
Any spherical coordinate triplet (or tuple) <math>(r,\theta,\varphi)</math> specifies a single point of three-dimensional space. On the reverse view, any single point has infinitely many equivalent spherical coordinates. That is, the user can add or subtract any number of full turns to the angular measures without changing the angles themselves, and therefore without changing the point. It is convenient in many contexts to use negative radial distances, the convention being <math>(-r,\theta,\varphi)</math>, which is equivalent to <math>(r,\theta{+}180^\circ,\varphi)</math> or <math>(r,90^\circ{-}\theta,\varphi{+}180^\circ)</math> for any {{mvar|r}}, {{mvar|θ}}, and {{mvar|φ}}. Moreover, <math>(r,-\theta,\varphi)</math> is equivalent to <math>(r,\theta,\varphi{+}180^\circ)</math>.

When necessary to define a unique set of spherical coordinates for each point, the user must restrict the [[interval (mathematics)|range, aka interval]], of each coordinate. A common choice is:

{{startplainlist|indent=1}}
* radial distance: {{math|''r'' ≥ 0,}}
* polar angle: {{math|0° ≤ ''θ'' ≤ 180°}}, or {{math|0 rad ≤ ''θ'' ≤ {{pi}} rad}},
* azimuth : {{math|0° ≤  ''φ'' < 360°}}, or {{math|0 rad ≤  ''φ'' < 2{{pi}} rad}}. 
{{endplainlist}}

But instead of the interval {{closed-open|0°, 360°}}, the azimuth {{mvar|φ}} is typically restricted to the [[interval (mathematics)#Definitions|half-open interval]] {{open-closed|−180°, +180°}}, or {{open-closed|−{{pi}}, +{{pi}} }} radians, which is the standard convention for geographic longitude.

For the polar angle {{mvar|θ}}, the range (interval) for inclination is {{closed-closed|0°, 180°}}, which is equivalent to elevation range (interval) {{closed-closed|−90°, +90°}}. In geography, the latitude is the elevation.

Even with these restrictions, if the polar angle (inclination) is 0° or 180°{{mdash}}elevation is  −90° or +90°{{mdash}}then the azimuth angle is arbitrary; and if {{mvar|r}} is zero, both azimuth and polar angles are arbitrary. To define the coordinates as unique, the user can assert the convention that (in these cases) the arbitrary coordinates are set to zero.

=== Plotting ===
To plot any dot from its spherical coordinates {{math|(''r'', ''θ'', ''φ'')}}, where {{mvar|θ}} is inclination, the user would: move {{mvar|r}} units from the origin in the zenith reference direction (z-axis); then rotate by the amount of the azimuth angle ({{mvar|φ}}) about the origin  ''from'' the designated ''azimuth reference'' direction, (i.e., either the x- or y-axis, see [[#Definition|Definition]], above); and then rotate ''from'' the z-axis by the amount of the {{mvar|θ}} angle.