The Ultimate 9-point conic document

[CrypticSpren he/him]

Been working on a mega doc for all things 9-point conic. It covers a number of things and I want to push this out to the wider Rithmatics community.

Sections 1.1-1.3: Making general information about 9-point conics accessible

9-point conics are central to Rithmatics, but it's pretty hard to find good documentation about them. I've literally only been able to find 4 sources yielding any information (besides the 17th shard) about 9-point conics. All of them... kind of suck

Spoiler

- Wikipedia: no proof

- Wiki source 1, the original paper: page only has statement of existence, no proofs https://www.jstor.org/stable/1967142?seq=1

- Wiki source 2, a scan of a paper from 1912 that is physically difficult to read because it's not typeset https://babel.hathitrust.org/cgi/pt?id=uc1.b3808276&view=1up&seq=1

- A fine arts thinkpiece that is miserable to read http://mat.msgsu.edu.tr/~dpierce/Mathematics/Geometry/Thales/nine-point-conic-2017-07-25.pdf

- A "proof by computer" that handwaves all 9-point conics as affine transformations of the 9-point circle... which I do not think is true for hyperbolas https://www.researchgate.net/publication/233284234_The_nine-point_conic_A_rediscovery_and_proof_by_computer

Most of the doc covers facts hardcore Rithmatic theorists already know from @KalynaAnne (namely what the 9-point conic is and the criterion for determining type of conic (circle, ellipse, or hyperbola) -- but actually has proofs for them.

I'm quite happy with how these turned out overall -- I haven't seen other sources do everything in terms of the centered conic matrix (they just use the raw conic formula), and I think using the matrix helps a lot in making things understandable.

I also have a section on what I'm calling the Conjugate Diameter Property, which I haven't seen anyone mention before. I think any aspiring Rithmatists will find the associated corollary (the Tangent-Direction Property) very useful for drawing defenses

Section 1.4-1.6: Classifications of sub-9 point configurations

Enumeration of the different n-point configurations and their properties, taken from the hardcore rithmatic theory thread. I also have what I think is a satisfying mathematical explanation that explains why the only valid 4-point and 2-point circles are the ones described in the book, despite being able to draw circles through any rectangle and in many ways through 2 points, that generalizes well to ellipses and hyperbolas.

Section 2: Inverse Algorithms

If I and the other suckers at the game thread ever get our act together and really create a game, we need to be able to take a collection of attachments to a Line of Warding and detect whether a corresponding quadrangle that admits the given conic exists. The algorithms here should cover all circle, ellipse, and hyperbola cases

Link to overleaf: https://www.overleaf.com/read/khgdzgrmnjzd#e3571f

9_Point_Conics.pdf

If anybody is interested in contributing, please let me know.

• The whole thing is pretty wordy and can probably be trimmed down; it's pretty long
• I'm the only person who has looked at the math. Though I think it looks good, a check for readability and correctness is very welcome
• I got tired of writing and don't have much covering parabolic and degenerate conic configurations -- which probably should be covered, if believers in Line of Forbiddance (and, as @IlstrawberrySeed has suggested on our mostly-dead discord server, the Mark's Cross) as an instance of a segment of a degenerate conic are correct -- is totally lacking in both the classification and inverse algorithm sections of the doc

 

Edited May 30 by CrypticSpren
Reupload without AI

[Carbonationspren he/him]

I am not qualified to check all the math, but you did successfully nerd-snipe me into creating this: https://www.desmos.com/calculator/p3plgnrdek

It's quite fun to fiddle with (though it gets a little confused on some of the degenerate cases because I'm using Desmos' regression feature instead of a closed-form solution) 

[CrypticSpren he/him]

Originally, unaware of the recent rule changes to the site, I had used a lot of AI assistance in generating the document.

Now I have completely rewritten it, without AI usage!

In terms of differences from the old document

Improvements

- Fixed arithmetic error in proof of 9-point conic

- Clean up of conjugate-diameter property symbol manipulation parts

- Complete overhaul of admissibility proof for 4-point conic (the previous one was handwavy at some parts and unnecessarily convoluted in others)

Dropped Portions

- Proof of "9-point conic is a circle if and only if one of the quadrangle points is at the orthocenter of the triangle)". I feel like I've seen a purely geometric proof that is satisfying for this out there somewhere. If I can't find it though I will add this section back.

- Handing of input error from algorithms section. I worry they may have obstructed clarity initially. For the most part readers can figure out the sensible thing to do by just replacing every instance of "A = B" with "A - B < epsilon" throughout the algorithms. The only notable difference besides this is that doing so flag some diagonals as midpoints incorrectly if someone draws a 9-pointer that is very close to a degenerate configuration -- but the best way to handle such cases is probably worth discussing in playtesting.

- Proofs of soundness from algorithms section. Hopefully it should be obvious without spelling it out explicitly

- Diagrams for all 9-point configurations. Tikz is pretty onerous to use and I just don't want to go through the hassle of writing scripts to generate it. Hopefully @KalynaAnne's tumblr posts and @Carbonationspren's wonderful Desmos graph should let you all figure out what the configurations look like yourself easily enough. If any readers care enough for better visualizations though, I'm not opposed to making some Desmos graphs or manim plots myself.

Edited May 30 by CrypticSpren
Rewrite to be relevant to current document status