A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2021 Jun 5, 17:15 -0700
That's an interesting question: could you modify the Snellius Construction to three dimensions? I agree with you that it seems like it would work. And what would a 3-D station pointer look like?? Four arms sticking up at various angles... and then you wiggle that around in your 3-D astrometrics lab until everything lines up properly. Heh. There's a better way.
One advantage of splitting the universe of objects into two groups, near stuff and far stuff, is that the far stuff serves as a background of pure directions, unlike the coastal navigation problem addressed by the Snellius construction where the objects in question are relatively close to the observer. If we can measure large angles, we can get by with a very small number of objects for our navigation: let's go with three distant stars, A, B, C, and two nearby stars, M and N. We assume that A, B, and C have known angular coordinates, RA and Dec, and very great distances while M and N have relatively well determined cartesian coordinates x, y, z in some arbitrary coordinate system.
We measure the angles from the three distant stars, A, B, C, to the first of our nearby stars, M. The angles could have any values from 0-180°. And what we end up with is the geometry of a standard three-body fix. Three circles will cross at one point (or in a tiny triangle), and that crossing point is a particular direction: RAM and DecM. Our observer must be sitting on a ray that emanates from the 3-D coordinates, x, y, z, of star M pointing towards a direction opposite to the direction of the crossing point (that is, the ray points toward RAM+180°, -DecM). In effect, this is a 3-D line of position aiming directly away from star M. We repeat this process with star N and the two LOPs (or really "rays" of position) will cross at a single point. That's your fix.
This process can be generalized to a large number of stars in both groups, and the statistics conspire to give us an accurate fix. For a given uncertainty in the angles measured, our LOPs are more accurate in direct proportion to the parallax of the bodies (nearer stars in the "near stuff" group perform better).
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