| Revision as of 18:20, 21 August 2003 edit61.102.142.86 (talk)No edit summary← Previous edit | Revision as of 04:15, 18 January 2004 edit undoTakanoha (talk | contribs)733 edits #REDIRECT coordinates (elementary mathematics)Next edit → | ||
| Line 1: | Line 1: | ||
| #REDIRECT ] | |||
| ]] | |||
| '''Polar coordinate systems''' are ]s 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 coordinate systems are listed below. | |||
| === Circular Polar Coordinates === | |||
| A two-dimensional coordinate system, defined by an origin, <i>O</i>, and a semi-infinite line <i>L</i> leading from this point. ''L'' is also called the polar axis. In terms of the ], one usually picks ''O'' to be the origin (0,0) and ''L'' to be the positive x-axis (the right half of the 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>). | |||
| === Cylindrical Polar Coordinates === | |||
| (Also see ]) | |||
| 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 P 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 orthogonal projection of the point P onto the XY plane. This is the same as the distance of P to the z-axis. | |||
| * <i>θ</i> is the angle between the positive x-axis and the 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>). | |||
| === Spherical Polar Coordinates === | |||
| (Also see ].) | |||
| 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 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. | |||
| 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:''' | |||
| *] | |||
Revision as of 04:15, 18 January 2004
Redirect to: