Polar coordinates: Difference between revisions

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	'''Polar coordinate systems''' are coordinate systems in which a point is identified by a distance from some fixed feature in space and one or more subtended ].



	The principal types of polar co-ordinate systems are:



	<i>Circular Polar Coordinates</i>



	A two-dimensional coordinate system, defined by an origin, <i>O</i>, and a semi-infinite line <i>L</i> leading from this point. (In terms of the ], the origin (0,0) and the positive x-axis).



	A point P is then located by its distance from the origin and the angle between line <i>L</i> and OP, measured anti-clockwise. The co-ordinates are typically denoted <i>r</i> and <i>θ</i> respectively: the point P is then (<i>r</i>, <i>θ</i>).



	<i>Cylindrical Polar Coordinates</i>



	A three-dimensional system which essentially extends circular polar coordinates by adding a third co-ordinate (usually denoted <i>h</i>) which measures the height of a point above the plane.



	A point is given as (<i>r</i>, <i>θ</i>, <i>h</i>). In terms of the Cartesian system:

	* <i>r</i> is the distance from O to P', the projection of the point P onto the XY plane,

	* <i>θ</i> is the angle between the positive x-axis and line OP', measured anti-clockwise

	* <i>h</i> is the same as <i>z</i>.

	Some mathematicians indeed use (<i>r</i>, <i>θ</i>, <i>z</i>), especially if working with both systems to, to emphasise this.



	<i>Spherical Polar Coordinates</i>



	This system is another way of extending the circular polar system to three dimensions, defined by a line in a plane and a line perpendicular to the plane. (The x-axis in the XY plane and the z-axis.)



	For a point P, the distance co-ordinate is the distance OP, not the projection. It is sometimes notated <i>r</i> but often <i>ρ</i> (Greek letter rho) is used to emphasise that it is in general different to the <i>r</i> of cylindrical co-ordinates.



	The remaining two co-ordinates are both angles: <i>θ</i> is the anti-clockwise between the x-axis and the line 0P', where P' is the projection of P in the XY-axis. The angle <i>φ</i>, measures the angle between the vertical line and the line OP.



	In this system, a point is then given as (<i>ρ</i>, <i>φ</i>, <i>θ</i>).



	Note that <i>r</i> = <i>ρ</i> only in the XY plane, that is when <i>φ</i>= <i>π</i>/2 or <i>h</i>=0.



⚫	~~:''See also:''~~ ]

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