Playing with Coordinate Systems (Spherical Geometry)

[solarblast]

I'm working on a problem that involves two Earth stations that scan the skies. I'm writing a simulation program (no physics involved) that simply finds the az/alt of an event observed simultaneously by each station. At this point, I'm warming up to the mathematics, spherical geo, etc. to pull this off. It's been awhile since I've dealt with this sort of thing. A long while. My approach was to start with lat, long coordinates and then move things to an xyz coordinate system, and do x,y, or z rotations as required. I'm using 3x3 matrices for those, and generally vectors.

Here's a simple example I'm working on. x is pointing south, y east and z through the north pole. I want to pick an arbitrary point and draw a circle of a radius in degrees around it on the earth.

Suppose I'm content with one point on the circle to make this easy, and let's take the set up along 0 longitude.

Center of circle: (0.0, 20.0) (long,lat)
Point on circle: (0.0, 30.0)
I want the point on the circle to be 90 deg ccw to the west.

I then do a few rotational transforms to work around the (0,0) lat/long point, and look at the results for where the point on the circle is located.

Seemingly, that should be at (-10,20) in the original xyz. But I get
(19.68,-10.63).

Since I'm drawing a circle (one point here) and it is not (after the "circle" position rotation) on a great circle, the results seem close enough. However, this inaccuracy suggests I'm going about this incorrectly. In fact, I'm pretty sure of it.

My suspicion is that I've got to ditch xyz, and work maybe entirely in spherical coordinates, and associated vectors, probably vectors something like (alpha, beta, r),where r is a radius, and the others are angles. In the above example, I decided to center everything around (0,0) long and lat. It probably should have been to center around the circle's center. Something like topocentric, but my skimpy knowledge of them suggests that's not such a good idea. Probably in spherical coords, I wouldn't really need rotations matrices.

Comments?

 

[Borg]

I'm pretty sure that spherical coordinates would be the best way to deal with this. However, it has been a long time since I wrote something like that either. From what I remember, you will need one set of equations to display things and another set to select something that is being displayed. Once you know the positions of the stations, the rest should fall into place.

 

[solarblast]

I think this is what I needed for finishing off matters <http://www.erikdeman.de/html/sail042e.htm>.