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Polar coordinates

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 coordinate systems are listed below.

Circular Polar Coordinates

A two-dimensional coordinate system, defined by an origin, O, and a semi-infinite line L leading from this point. L is also called the polar axis. In terms of the Cartesian coordinate system, 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 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

(Also see cylindrical coordinate system)

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 P is given as (r, θ, h). In terms of the Cartesian system:

• r 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.
• θ is the angle between the positive x-axis and the line OP', measured anti-clockwise.
• h is the same as z.
Some mathematicians indeed use (r, θ, z).

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 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 r but often ρ (Greek letter rho) is used to emphasise that it is in general different to the r of cylindrical co-ordinates.

The remaining two co-ordinates are both angles: θ 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 φ, measures the angle between the vertical line and the line OP.

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

Note that r = ρ only in the XY plane, that is when φ= &pi/2 or h

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