Editing Polar coordinate system (section)

==Conventions==
[[Image:Polar graph paper.svg|thumb|A polar grid with several angles, increasing in counterclockwise orientation and labelled in degrees]]
The radial coordinate is often denoted by ''r'' or [[rho|''ρ'']], and the angular coordinate by [[phi|''φ'']], [[theta|''θ'']], or ''t''.  The angular coordinate is specified as ''φ'' by [[International Organization for Standardization|ISO]] standard [[ISO 31-11|31-11]], now [[ISO/IEC 80000|80000-2:2019]]. However, in mathematical literature the angle is often denoted by &theta; instead.

Angles in polar notation are generally expressed in either [[degree (angle)|degree]]s or [[radian]]s (2[[pi|{{pi}}]] rad being equal to 360°). Degrees are traditionally used in [[navigation]], [[surveying]], and many applied disciplines, while radians are more common in mathematics and mathematical [[physics]].<ref>{{Cite book |last1=Serway |first1=Raymond A. |title=Principles of Physics |last2=Jewett Jr., John W. |publisher=Brooks/Cole—Thomson Learning |year=2005 |isbn=0-534-49143-X}}</ref>

The angle ''φ'' is defined to start at 0° from a ''reference direction'', and to increase for rotations in either [[clockwise|clockwise (cw)]] or counterclockwise (ccw) orientation. For example, in mathematics, the reference direction is usually drawn as a [[ray (geometry)|ray]] from the pole horizontally to the right, and the polar angle increases to positive angles for ccw rotations, whereas in navigation ([[bearing (navigation)|bearing]], [[heading (navigation)|heading]]) the 0°-heading is drawn vertically upwards and the angle increases for cw rotations. The polar angles decrease towards negative values for rotations in the respectively opposite orientations.

===Uniqueness of polar coordinates===
Adding any number of full [[turn (geometry)|turns]] (360°) to the angular coordinate does not change the corresponding direction.  Similarly, any polar coordinate is identical to the coordinate with the negative radial component and the opposite direction (adding 180° to the polar angle). Therefore, the same point (''r'', ''φ'') can be expressed with an infinite number of different polar coordinates {{nowrap|(''r'', ''φ'' + ''n'' × 360°)}} and {{nowrap|(−''r'', ''φ'' + 180° + ''n'' × 360°) {{=}} (−''r'', ''φ'' + (2''n'' + 1) × 180°)}}, where ''n'' is an arbitrary [[integer]].<ref>{{Cite web |date=2006-04-13 |title=Polar Coordinates and Graphing |url=http://campuses.fortbendisd.com/campuses/documents/Teacher/2012%5Cteacher_20120507_1147.pdf |url-status=dead |archive-url=https://web.archive.org/web/20160822034840/http://campuses.fortbendisd.com/campuses/documents/Teacher/2012%5Cteacher_20120507_1147.pdf |archive-date=August 22, 2016 |access-date=2006-09-22}}</ref> Moreover, the pole itself can be expressed as (0,&nbsp;''φ'') for any angle ''φ''.<ref>{{Cite book |last1=Lee |first1=Theodore |title=Precalculus: With Unit-Circle Trigonometry |last2=David Cohen |last3=David Sklar |publisher=Thomson Brooks/Cole |year=2005 |isbn=0-534-40230-5 |edition=Fourth}}</ref>

Where a unique representation is needed for any point besides the pole, it is usual to limit ''r'' to positive numbers ({{nowrap|''r'' > 0}}) and ''φ'' to either the [[interval (mathematics)|interval]] {{Closed-open|0, 360°}} or the interval {{Open-closed|−180°, 180°}}, which in radians are {{closed-open|0, 2π}} or {{open-closed|−π, π}}.<ref>{{Cite book |last1=Stewart |first1=Ian |title=Complex Analysis (the Hitchhiker's Guide to the Plane) |last2=David Tall |publisher=Cambridge University Press |year=1983 |isbn=0-521-28763-4}}</ref> Another convention, in reference to the usual [[codomain]] of the [[inverse trigonometric functions|arctan function]], is to allow for arbitrary nonzero real values of the radial component and restrict the polar angle to {{open-closed|−90°,{{nbsp}}90°}}. In all cases a unique azimuth for the pole (''r'' = 0) must be chosen, e.g., ''φ''&nbsp;=&nbsp;0.