[Proj] Geocentric vs. Geodetic latitude

[Paul Selormey]

Thanks for making a long story short.

Best regards,
Paul.

----- Original Message ----- 
From: "Glynn Clements" <glynn.clements at virgin.net>
To: <proj at remotesensing.org>
Sent: Thursday, April 29, 2004 12:39 AM
Subject: Re: [Proj] Geocentric vs. Geodetic latitude


>
> 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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