[Proj] Geocentric vs. Geodetic latitude
Glynn Clements
glynn.clements at virgin.net
Wed Apr 28 11:39:27 EDT 2004
Frank Warmerdam wrote:
> I was recently suprised to encounter locations identified as being
> geocentric that were give as a latitude and longitude. I had expected them
> to be x/y/z values in meters.
>
> A client of mine, better versed in such things explained that there is in
> fact a distinction between geocentric and geodetic lat/long. He wrote:
>
> >"The angle L' is called "geocentric latitude" and is defined as the
> > angle between the equatorial plane and the radius from the geocenter.
> >
> > The angle L is called "geodetic latitude" and is defined as the angle
> > between the equatorial plane and the normal to the surface of the
> > ellipsoid. The word "latitude" usually means geodetic latitude. This
> > is the basis for most of the maps and charts we use. The normal to the
> > surface is the direction that a plumb bob would hang were it not for
> > local anomalies in the earth's gravitational field."
>
> I need to implement some code to convert geocentric latitude to geodetic
> latitude. I think I can do from the above description, but I am wondering
> if anyone can confirm the above description, provide a forumla or most
> importantly provide some sample latitudes in both systems I can check my
> work against.
Given the parametric equation for an ellipse:
x = a.cos(t)
y = b.sin(t)
the tangent vector is:
dx/dt = -a.sin(t)
dy/dt = b.cos(t)
and thus the outward normal is:
nx = b.cos(t)
ny = a.sin(t)
By the above definitions, the geocentric latitude L' is given by:
tan(L') = y / x
= b.sin(t) / a.cos(t)
= (b/a).tan(t)
while the geodetic latitude L is given by:
tan(L) = ny / nx
= a.sin(t) / b.cos(t)
= (a/b).tan(t)
Thus the ratio of the two is:
tan(L) / tan(L')
= (a/b).tan(t) / (b/a).tan(t)
= (a/b) / (b/a)
= a²/b²
So:
L = atan((a²/b²).tan(L'))
and:
L' = atan((b²/a²).tan(L))
where a is the equatorial axis and b the polar axis.
--
Glynn Clements <glynn.clements at virgin.net>
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