This paper is concerned with justifying the claim/proposition that the asteroid can be considered the hyperbolic element of the 1-manifold version of the geometric lens. The paper is, essentially, one big proof of that proposition. Right now (8:52pm, 9/11/19), I'm kind of exhausted and I just want to go home and eat, so I'm just going to post the paper and go home. Future updates (including errata, I'm sure) to come.
You'll possibly have noticed, if you've read previous papers, thatI tried something....stylistically...different with this paper. I put all the definitions I was using in the paper in one section (section 1) as more of a reference. I felt it reads easier than breaking up the material/proof of section 2 with sporadic "now define X as blahblahblah", especially considering that most of the objects needing to be defined are fairly elementary and well-known. You go into the second section having everything defined already, and you can flip back if you need to be reminded. If I thought it was important that the reader be reminded of the definition, I wrote "as given in def 1.z".
Other than that......enjoy!!!
Update, 9/13/19....let the errata'ing begin!
ERRATA:
1) The address for the final source (number 8) got messed up because LaTeX gonna LaTeX. The correct address should be: https://proofwiki.org/wiki/Curve_is_Involute_of_Evolute. (I'm a fan and supporter of proofwiki. I think it's a useful tool and I think having a place where you can collect and showcase proofs and alternate proofs is a great thing for math in general).
2) This is less a correction than an aside. While I was writing this paper, it occurred to me that, in the 1-manifold/2-dimensional case at least, you can construct gL via the hypocycloid function. (see "a/b=4" case & its animation at http://mathworld.wolfram.com/Hypocycloid.html ) Notice that the path of the centre of the circle of radius b is exactly the path that would otherwise describe sq'(\star) in the 1-manifold/2-dimensional case. As of now, this fact doesn't really do much for constructions in dimensions higher than 2, but I thought it was a neat little bit of info/trivia none-the-less. Also, the hypocycloid function proves that the set of normal vectors of As one-to-one and onto wrt the circle (so this is pertinent to the "aligning normals" paper from last year). This hypocycloid construction also could help in making a MUCH more manageable function to describe sq'. Or not. I'll probably mess around with it and see what happens. I'll keep you posted, promise.
(Update 18 Sept. 2020: It occurred to me that parameterization of higher dimensional analogues can be obtained by taking the cross product of the hypocycloid function. So, for example, the parameterization of the 2-manifold As can be obtained by taking the cross product of 2 hypocycloids.)
3) I forgot the table of notations I normally attach. I mentioned in the introduction there would be a table of notations, too. Most notations are covered in Sect. 1, which also gives definitions. You can also check out the table of notations in my other papers related to gL stuff (basically, every paper except the economics one and the 3X3 magic square of squares one). None of the notation has really changed from those papers, except to be more precise (e.g.; in later papers I've added superscripts to things like "gL" and "As" to denote their manifold dimension).
4) I don't know why I'm just now realizing/noting this, but I've been using "t" where the more common notation would be "theta" since like the second (or third?) paper I wrote. I mean, it still holds the same meaning and I don't doubt that anyone could pick up the meaning, since I'm referring to it specifically as the polar variable (as opposed to, for example, the ̶E̶u̶c̶l̶i̶d̶e̶a̶n̶ CARTESIAN variable(s)). I just like to be complete with my corrections and clarifications, I guess.
5) In the "dumb mistakes I keep making" category of errata entries (of this and previous papers): I keep saying "Euclidean" when I mean "Cartesian" in the papers and associated errata. I swear to FSM that I kept telling myself "REMEMBER TO SAY CARTESIAN AND NOT EUCLIDEAN WHEN YOU MEAN CARTESIAN!!!!" in mental-all-caps (with multiple mental exclamation points) before writing this paper. And I keep making the same mistake despite my many reminders to myself (in fact, see entry #4 directly above this).
The worst part is, sometimes I really do mean Euclidean....so it muddles things even more. I will assume any mathematician with a working knowledge above....errrmmmm....high school geometry.... will easily spot and mentally correct my mistake, and will also be able to differentiate when I mean Euclidean and when I mean Cartesian. They will probably mentally and physically have a good laugh, too. Which I totally understand because it really is an A+ level Bonehead Error.
So one of the "long game" goals (and there are, indeed, multiple "long game" goals) of this project that I keep close to the chest (until now, I guess...though I'm still going to be purposefully vague) is that I'm trying to make a method to compare the way different structures behave in terms of their individual "inner structures" (e.g.: changing co-ordinate systems/variables in Euclidean space) and in terms of how the individual "inner structures" of one space compares with another's (e.g.; the differences and similarities in changing co-ordinate systems in Euclidean space VS. changing co-ordinate systems in Hyperbolic space). The method (obviously?) draws from geometry and topology, the intersection of the two subjects, and (so far) some smatterings of vague algebra.
[I realize that there are probably other methods of doing what I'm trying to do, but to the best of my knowledge using the n-sphere, the n-astroid/hyperbolic octahedron, and the n-cube as representatives for positive, negative, and 0 curvature, respectively, is....novel if not unique. Since these manifolds are basically ubiquitous in all metric spaces, and dimensions, they are (in my opinion) ripe for exploitation for this kind of study.]
In this light, my "Euclidean-Cartesian mix-up" errors are even more egregious.
Errata Update 6/16/2020::
This is more of a general thing concerning this paper AND all previous papers, but it occurs to me that I've been limiting myself and these manifolds (and their union) to specific ambient spaces (e.g.; 1-manifolds restricted to/embedded in 2 dimensional space). I'm more concerned with the more general case of, for n>0, the n-manifold gL^{n}(\star) (and it's component parts) that can be embedded in a general space. This could be either on it's own, or as a sub-manifold. Anyway, I thought it important to make the distinction here. I plan on treating the manifold(s) as the more general case in future papers, and will (eventually) go back and edit previous papers.