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| The principal types of polar co-ordinate systems are: | |||
| Subtypes include: | |||
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| A two-dimensional coordinate system |
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 Cartesian terms, 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>). | |||
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| A three-dimensional system |
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' | |||
| * <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. | |||
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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 it. (The XY plane and the z-axis.) | |||
| ''Needs to be filled in'' | |||
| 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. (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.) | |||
| The remaining two co-ordinates are both angles: <i>θ</i> is identical to its 2-dimensional counterpart and <i>φ</i>, which 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>). | |||
Revision as of 18:57, 19 January 2002
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 angles.
The principal types of polar co-ordinate systems are:
Circular Polar Coordinates
A two-dimensional coordinate system, defined by an origin, O, and a semi-infinite line L leading from this point. (In Cartesian terms, 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 L and OP, measured anti-clockwise. The co-ordinates are typically denoted r and θ respectively: the point P is then (r, θ).
Cylindrical Polar Coordinates
A three-dimensional system which essentially extends circular polar coordinates by adding a third co-ordinate (usually denoted h) which measures the height of a point above the plane.
A point is given as (r, θ, h). In terms of the Cartesian system:
- r is the distance from O to P', the projection of the point P onto the XY plane,
- θ is the angle between the positive x-axis and line OP'
- h is the same as z.
Some mathematicians indeed use (r, θ, z), especially if working with both systems to, to emphasise this.
Spherical Polar Coordinates
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 it. (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 r but often ρ (Greek letter rho) is used to emphasise that it is in general different to the r of cylindrical co-ordinates. (Note that r = ρ only in the XY plane, that is when φ= π/2 or h=0.)
The remaining two co-ordinates are both angles: θ is identical to its 2-dimensional counterpart and φ, which measures the angle between the vertical line and the line OP.
In this system, a point is then given as (ρ, φ, θ).
- See also: Cartesian coordinate system