Spherical Coordinates in Three Dimensions
A point in three dimensional space is designated by spherical coordinates in the following way:
r: |
|
the distance from the origin |
q: |
|
the angle rotated from the positive xaxis (azimuth) |
f: |
|
the angle rotated from the positive zaxis (declination) |
The point P in the figures below is the point designated by r = ,
, and
(the cartesian coordinate (2, 2, 3)). The coordinate transformations are the following:
Cone and plane
Cone , plane , and sphere r =
The figures above depict the level surfaces for spherical coordinates. The surface r = c, (c > 0) is a sphere of radius c since r is the distance a point is from the origin. So we are looking at all the points that are the same distance from the origin. The surface = c is a plane containing the zaxis similar to cylindrical coordinates. The surface f = c defines all the points that lie on a line rotated an angle of c radians from the positive zaxis. If we allow r and q to take on any values the surface generated is a cone. The figures above also indicate how a point is located using spherical coordinates. Move from the origin on the cone given by f until a distance r from the origin, all of this is done directly above the xaxis.
Back