Re: simple geometry problem

On Wed, 09 Aug 2006 03:21:28 -0700, pieterprovoost wrote:

Hi,

This is the situation: for a certain position on earth (p1) I have
calculated the xyz coordinates and I have constructed a plane (A1)
through that point and perpendicular to the earth radius going trough
it (this plane is how the observer perceives the earth's surface). I
then calculated the point where this plane intersects the earth axis
(a1). The vector from p1 to a1 (v1) is the north direction for an
observer in p1 (on the northern hemisphere of course). I now want to
calculate the vector in the plane A1 which has a certain angle (alpha1)
with v1. How can I do this?

Thanks
Piet
A common convention is to use a (left handed) "topocentric" coordinate
system U (north) V (east) and W (up). At a point with latitude phi
and longitude lambda (in radians), the topocentric coordinates are related
to the (right handed) geocentric cartesian coordinates X,Y,Z by
(U) = P2 * R2(phi-pi/2) *R3( lambda-pi)* (X)
(V) (Y)
(W) (Z)
where P2 is
( 1 0 0 )
( 0 -1 0)
( 0 0 1)
R2 is rotation about the y axis and R3 rotation about the
z axis
In the UVW coordinates your vector is (cos(alpha1), sin(alpha1), 0)
To get the XYZ coordinates of this you want to apply to this the
inverse of the above transformation, ie
R3(pi-lambda)*R2(pi/2-phi)*P2

Duncan
PS Exactly the same formulae hold if you model the earth
as an ellipsoid rather than a sphere, as long as you take the
latitude to be the geodetic latitude (the angle between the local
vertical and the plane of the equator) rather then the geocentric
latitude (the angle between the line from the centre of the earth to
the point and the plane of the equator).

.

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