It turns out to be very convenient for the problem solving in cartography to operate with the so called spherical co-ordinates instead of cartesian co-ordinates (figure from [3] under literature).
Thus any point P in space can be determined by the number triple (R; v; u) instead of the number triple (x; y; z). Here it is:
R = radius of sphere = distance between the point P and the origin O ( 0 ≤ R ),
v = angle between the line OP and the xy-plane ( - / 2 ≤ v ≤
/ 2 ) and
u = angle between the projection of the line OP onto the xy-plane and the positive x-axis ( - ≤ u ≤
).
The illustration shows the rotational direction of the goniometry. The values R, v, u are named spherical co-ordinates of the point P. They correspond to the polar co-ordinates in the plane and therefore are also called spacial polar co-ordinates.
Any triple of spherical co-ordinates accords to exactly one point in space. Contrariwise a point P in space accords to a single triple of spherical co-ordinates only if P does not lie onto the z-axis: On the z-axis without the origin O only R and v (± / 2) are unique, in contrast u is arbitrarily. If P is equal to the origin O, thus only R = 0 is unique, v and u are arbitrary.
See also | |
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parameter representation |
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parameter lines |