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In a prior question about the altitude of an irregular triangular pyramid, I got two good answers that solved the problem but I don't understand the concepts as well as I think I should. Can anyone recommend one or more sources from which I can learn how the equations were developed? I really like real books, by the way, and I'm willing to buy more than one or two if that's what it takes for me to learn 3-dimensional trigonometry in depth.

poetasis
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    https://archive.org/details/atreatiseonplan00hobsgoog/page/n6 https://archive.org/details/planetrigonomet02lonegoog/page/n8 – lab bhattacharjee Oct 18 '18 at 17:20
  • These links are good but I don't think they will help me understand the equations used in finding the height of an irregular triangular pyramid because the tittles refer to plane trigonometry. – poetasis Oct 18 '18 at 18:36

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Trigonometry is an inherently planar tool. If you want to be able to solve problems in geometry in $\mathbb{R}^3$, then you want to look at vector and matrix algebra, and maybe some multivariate calculus. That should give you the tools to solve problems such as the one you linked.

Alex Jones
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  • I can solve simultaneous equations by several means. I just want to know how to develop the equations shown in my prior post as correct. They work but I want to know why they work. – poetasis Oct 18 '18 at 19:48
  • There's some amount of geometric intuition that you need to develop to come up with those equations. For example, I think Narlin's answer is the most natural one. You can easily find the 3 points of the base (this is your standard "planar" trig). Once you have that, you have 3 points and 3 distances that all end at the same point. This is equivalent to the intersection of 3 spheres (where the points are the centers, and the distances are the radii). That sets up your equations for you. This isn't something that's necessarily covered in a book, it's just geometric intuition. – Alex Jones Oct 18 '18 at 19:57
  • Finding the coordinates of the base is the easy part. I don't have or want CAS etc. I want to know how to do this by hand or in spreadsheet or in a procedural language. I agree with what you say but I know very little about the computations needed to use three spheres to find a point of intersection of their radii. Soooo,.. I need to get up to speed of spherical trig or something. I learn well from books. Any recommendations? – poetasis Oct 19 '18 at 16:48
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    It sounds like geometry is where your weakness lies. In order to remedy this, I might suggest Euclid's Elements (or rather some modernization thereof), where book 11 to 13 cover 3D geometry based on the work of the first 10 books. It can be read with absolutely no background, if you start from the beginning. If you're very comfortable with rigorous linear algebra, then Audin's Geometry might help you. It provides a more algebraic approach to affine and projective geometry, and chapter VI on quadrics (of which the sphere is one) might be of help. It is a much more difficult read though. – Alex Jones Oct 19 '18 at 18:48
  • I have Euclid's Elements but I didn't think to look there because it doesn't read as well as modern texts. I also have Principia Methematica (both Newton's and Whitehead's). I will look at those and I will order Audin's Geometry. Let me know if you think of other books that would help. I will also look into the recommendations in your answer. – poetasis Oct 19 '18 at 23:21
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    Audin's Geometry should arrive in the next two weeks. I just with it were available in hardcover. No big deal, just a preference for my library. Thanks. – poetasis Oct 20 '18 at 09:27