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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 ]. |
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{{Redirect category shell| |
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The principal types of polar co-ordinate systems are listed below. |
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{{R with history}} |
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{{R from related word}} |
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=== Circular Polar Coordinates === |
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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). In other words, this axis (called the polar axis) is like the right half of the x-axis. |
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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>). |
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=== Cylindrical Polar Coordinates === |
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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. |
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A point is given as (<i>r</i>, <i>θ</i>, <i>h</i>). In terms of the Cartesian system: |
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* <i>r</i> is the distance from O to P', the projection of the point P onto the XY plane, |
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* <i>θ</i> is the angle between the positive x-axis and line OP', measured anti-clockwise |
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* <i>h</i> is the same as <i>z</i>. |
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Some mathematicians indeed use (<i>r</i>, <i>θ</i>, <i>z</i>), especially if working with both systems to, to emphasise this. |
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=== Spherical Polar Coordinates === |
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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.) |
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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. |
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The remaining two co-ordinates are both angles: <i>θ</i> is the anti-clockwise between the x-axis and the line OP', 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. |
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In this system, a point is then given as (<i>ρ</i>, <i>φ</i>, <i>θ</i>). |
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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. |
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'''See also:''' |
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