Editing Spherical coordinate system (section)

=== 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.