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Coordinate Systems

Up until now we have been representing a point as an ordered triplet (x, y, z) by specifying the distances along the x, y, and z axes, but often, it can be more useful to define the point using a different coordinate system.

Spherical Polars

Suppose we are in a satellite, orbiting some planet and we want to transmit our coordinates to someone. We could do this by first giving our distance from the centre of the planet, r, - this would confine our possible position to a sphere radius r centred at the centre of the planet. If we then give the longitude and latitude on the surface of the planet of the point directly below us, we have fully given our position - the intersection of the sphere of radius r, with the line joining the centre of the planet to the point directly beneath us.
These are spherical coordinates r, q, and f :

For point A at (r, q, f )
r is the distance of A from the origin, 0O r
q is the angle between a and k, ie. the angle made with the z axis.
Images the plane that contains the z axis and the point A, then
f is the angle between this plane and the xz plane.
That's it - given x, y, z we can find r,q, f and vice versa:
From trigonometry we can see that,
x = r sin q cos f
y = r sin q sin f
z = r cos q
and,
r = e ( x2 + y2 + z2) 0O r
q = cos-1 ( z/ e ( x2 + y2 + z2) ) 0 O q O p
f = tan-1 ( y / x ) + np 0 O fO 2p
where,
n = 0 if x > 0,
n = 1 if x < 0 and y O 0
n = -1 if x < 0 and y < 0
If x = 0 you can see by inspection whether f = p/2 or 3p/2

Cylindrical Polars


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