# Talk:Quadrisecant

This review is transcluded from Talk:Quadrisecant/GA1. The edit link for this section can be used to add comments to the review.
Reviewer: Ovinus (talk · contribs) 19:52, 24 July 2022 (UTC)
Alright. Haven't heard of this topic before.

I'll see if I can make a prettier image. Maybe even an STL would be cool, since a reader can "pivot" around to see the quadrisecants and infinite trisecants.
Thinking about how to do this now—should be fun. You are the computational geometer here, so I was wondering if you had any suggestions for finding quadrisecants numerically. I'm thinking a straightforward approach by choosing a random secant, perturbing it into a trisecant (within some epsilon), and then perturbing that into a quadrisecant. That should be at most quadratic time in the number of samples, per random initial choice. Ovinus (talk) 20:11, 26 July 2022 (UTC)
Finding the quadrisecants first and building the curve around them seems easiest to me. —David Eppstein (talk) 03:55, 27 July 2022 (UTC)
That is clever. I'll try that if I can't figure out my original idea; I want the knot to deform smoothly so that the reader can see how the quadrisecants move continuously. Anyway, I'm quite busy irl at the moment. Will put stuff on talk if/when I come up with anything. Ovinus (talk) 08:11, 27 July 2022 (UTC)
"However, quadrisecants are especially relevant" Are you contrasting quadrisecants with plain secants and trisecants? If so I'd be explicit, like "Compared to secants and trisecants" or something. "However" implies a weak contradiction, the nature of which is not obvious
Will continue in a bit. Ovinus (talk) 19:52, 24 July 2022 (UTC)

Thanks! A better image would be welcome; at least the present one is clearer than the one that was in there before. I changed the wording as you suggested above. —David Eppstein (talk) 20:04, 24 July 2022 (UTC)
LOL! I quite nearly spit out my water at that. Ovinus (talk) 20:21, 24 July 2022 (UTC)
Continuing:

"In contrast, if an arbitrary space curve is perturbed by a small distance to make it generic, there will be no lines through five or more points of the perturbed curve." Anything about degrees of freedom here?
"Degrees of freedom" and "generic curve" are not compatible concepts. The point of being generic is that it is not restricted to a class with finitely many degrees of freedom.
Ah, I should have been more clear. As in, an easy-to-understand heuristic explanation for why four points is discrete in the generic case, while five points requires special circumstances, based on the degrees of freedom of four points on a knot vs. of lines in 3D space. Perhaps that is misguided. In other words: is there a broadly understandable explanation for why 4? Why are trisecants not also discrete; why are "quintisecants" not a general occurrence? Ovinus (talk) 01:37, 27 July 2022 (UTC)
Ok, I found and added a visual explanation involving degrees of freedom from Wall. —David Eppstein (talk) 03:55, 27 July 2022 (UTC)
"Additionally, for generic space curves, the quadrisecants form a discrete set of lines that, in many cases, is finite" Are there any cases where it's infinite for a generic curve? I can't immediately think of one.
Take any system of countably many lines and any system of four points per line, and connect the dots by a generic curve. —David Eppstein (talk) 23:41, 26 July 2022 (UTC)
Interesting. Maybe put in a footnote? Ovinus (talk) 01:37, 27 July 2022 (UTC)
I added a line later about a conjecture that every wild knot has infinitely many. —David Eppstein (talk) 03:35, 27 July 2022 (UTC)
Exciting. Ovinus (talk) 08:11, 27 July 2022 (UTC)
"locally flat" Link to local flatness
You seem to have already made this minor edit. —David Eppstein (talk) 23:41, 26 July 2022 (UTC)
Oops! Ovinus (talk) 01:37, 27 July 2022 (UTC)
"In spaces with complex number coordinates rather than real coordinates" The jump to complex numbers is a bit sudden; it's no longer a plain old space curve
Well, it really kind of is. It lives in a space with three coordinates and three dimensions. Just one built on a different field. Anyway, I'm not sure what you think should be different here. —David Eppstein (talk) 03:56, 27 July 2022 (UTC)
I was thinking about this in conjunction with the next point, so that it's clear what collinearity means in a complex space, but I think it's straightforward enough to those with the background to fully appreciate the statement. All good. Ovinus (talk) 08:11, 27 July 2022 (UTC)
It'd be nice somewhere to see how quadrisecants can be put into symbols, even if it's obvious to some people. "Any knot may be parametrized as a function 
  
    
      
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    {\displaystyle f:[0,1]\to \mathbb {R} ^{3}}
  
 ... A quadrisecant is thus determined by any set of distinct inputs 
  
    
      
        
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    {\displaystyle t_{1},t_{2},t_{3},t_{4}}
  
 for which their images are collinear." Just something like that.
I didn't find any sources talking about quadrisecants of parametric curves. But how is "the images of distinct inputs 
  
    
      
        
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    {\displaystyle t_{1},t_{2},t_{3},t_{4}}
  
 of a curve parameterized by 
  
    
      
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    {\displaystyle (f_{x}(t),f_{y}(t),f_{z}(t))}
  
 simpler than "four distinct points on a space curve"? What is gained by considering the parameterization? —David Eppstein (talk) 05:29, 27 July 2022 (UTC)
That's fair, since many of these curves are determined implicitly Ovinus (talk) 08:11, 27 July 2022 (UTC)
"If five lines ... " What restrictions must be imposed on these lines? Need they be distinct, skew, etc
As the article states: "all five are intersected by a common line but are otherwise in general position". General position implies pairwise distinct and skew. —David Eppstein (talk) 05:22, 27 July 2022 (UTC)
Ah okay. Glossed over the second part. Ovinus (talk) 08:11, 27 July 2022 (UTC)
"If the fourth of the given lines pierces this surface, its two points of intersection" Why is it guaranteed to have two points of intersection in general?
Because it's a surface defined from a quadratic equation. The restriction of that quadratic to a line is still a quadratic and has two roots. —David Eppstein (talk) 03:57, 27 July 2022 (UTC)
That makes sense. Thanks for clarifying in the article. Ovinus (talk) 08:11, 27 July 2022 (UTC)
That's all for now. Ovinus (talk) 20:11, 26 July 2022 (UTC)

I think I'm happy with the article. Will pass tomorrow after one more readthrough. Ovinus (talk) 08:11, 27 July 2022 (UTC)