I'm trying to understand the analytic geometry (AG) aspects of a 1987 paper by Ceplecha that calculates the trajectory of a meteoroid (a rock) in the earth's atmosphere.
Rather than dealing with the complexities of his paper, I'd like to start simply with an article he wrote in a book called Exercises in Astronomy. It is posed as a problem for students. The article (problem) is titled Atmospheric Trajectory of a Meteoroid. I would think anyone well familiar with AG, should be able to understand the crux of the problem described there that I'm interested in. Even though it's been many, many years since I studied the subject, I have a pretty fair idea of how it works, but would like some assistance.
I'll start simply. Presently, my hang up is confined to problem (1), which is on pages 58 and 59. About 8 lines from the top of page 58 he writes "...system of parallel lines ..." Somehow this seems of no use. He's already defined the two planes and their intersection.
Let me provide some back ground about figure 26. It shows two observing stations, A and B, possibly separated by say 50 to 100 miles. A fireball (bright meteor produced by a meteoroid (the rock) is burning in the earth's atmosphere and is moving some distance above the stations across the sky. The plane of the meteoroid (physics) provides the meteoroid will be on a great circle. Each station has photographed the meteor with a video camera and has many seconds of data on frames of 1/30 seconds duration. The large plane showing N and W arrows is tangent to the earth's surface 1/2 the way between A and B. It is often the case one station will not see the same start and end time of the other. However, time doesn't really play into the problem posed here. The zenith distance is the vertical height of the meteoroid above the big plane drawn. Although in it's full complexity, 1987 Ceplecha, all points are used to determine the plane of travel, he simplifies things by using on the first and last point seen by each observer. Note that for purposes here, he's assuming the meteoroid is on a straight line.
He poses (1) as a graphical problem, but at this point I'm curious about his references to parallel lines has to any of this. He clearly states that the solution is the intersection of the two planes he mentions. Consider (1), but the interesting parts a (3) and (4).
Note the missing page 61 and possibly 62 (I happen to have a copy of 62 directly from the book) are astronomical formulae useful in computing, for example, the radiant, which is the point in the sky where the meteoroid appears to come from.
If anyone has questions, I'll try to answer them.
If I can get by this article, I think I have a better chance of understanding his 1987 article, which gets into both linear (LSQ) and non-linear squares (NLSQ) to find the intersection of the two planes, and extends the idea more than two stations. I have the barest of clues on how he treats the multi-station solution.