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    Re: Symmedian for a tetrahedron
    From: Frank Reed
    Date: 2017 Feb 5, 17:00 -0800

    David P., You wrote:
    "Do you know if the tetrahedron can be shrunk away completely by slewing the GNSS receiver clock"

    I don't believe Bill Lionheart was referring specifically to a GNSS case. He was simply describing something that could exist in three-dimensional space in a general case. For example, if you shot angles between stars and celestial bodies (which are effectively "lunars in space", but at known UT), then you would get a cone of position from each sight, and locally those cones are plane surfaces. Four such plane surface would cross at a point if there were no error in the observations, but in the general case they will cross in a small tetrahedron analogous to the cocked hat of surface celestial. One approach would be to treat the size of the tetrahedron as a systematic error. Then the tetrahedron could be reduced to zero by adding some offset to each observation (this is quite similar to the GNSS case). But that would be wrong-headed if the error is random. If the error is random then, similar to the case on the Earth's surface, the most probable position would be at the "symmedian" for the tetrahedron. In some cultures of navigation in the late 20th century, this same error was made in surface navigation; navigators were taught that they could reduce the size of the "cocked hat" by applying a systematic error. This created the illusion of an improved fix. The technique is valid only when we have an a priori reason to believe that there is some systematic error (maybe we didn't measure dip?). As in the three-dimensional case of the tetrahedron, if we assume that there is no systematic error, then we can't collapse the triangle and the most probable position is then at the symmedian point of the triangle.

    In the real world, there is probably very little use for this bit of geometry about the three-dimensional symmedian point. In satellite and spacecraft navigation, there has probably never been a case of pure celestial navigation --where angles produce cones of position, and crossing cones of position directly yield a position fix. Astronomical sights are combined with a slew of other navigational information like radio range and rate data. These disparate sorts of data are all incorporated into the bext fix usually using a type of weighted average, a statistical algorithm known as a "Kalman filter" (that short description nearly exhausts my knowledge of this technique!). Additionally, the observations do not constrain the position of the satellite or spacecraft so much as its state vector, which is basically a case of orbit determination. Position is part of this process, but the optimization is done in a higher-dimensional mathematical space since the motion is held beautifully to a limited set of trajectories and orbits by the law of universal gravitation.

    Frank Reed

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