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Polar coordinate system
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===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, ''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'' = 0.
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