Can the hive mind help?
I have a nice, short proof for a theorem about conics. Unfortunately, I don't know how well the theorem itself is known. I'm sure it can be found in some book written in the last 200 years, but... 🤷
Does anyone know the following statement?
Given two non-degenerate conics in the projective plane. There are four lines which are tangent to both conics. They touch the given conics in four points each. Then there exists another conic, which runs through these eight points.
Ha! As soon as I told someone (you) about it, I found it in a book. (That happens way too often...)
The theorem can be found in a more powerful form in Hatton's "Principles in Projective Geometry" from 1913. Theorem 133 on the left page in the attached image.
(It's the dual of mine, but I prefer the final conic to be a conic through points.)
@ColinTheMathmo, would mind sending this to your mailing list, and thank them for their help in my regard?
@BernhardWerner Would you like me to ask on some other networks? Or would you prefer I asked a few individuals?
@BernhardWerner OK, I've asked the "Math Fun" mailing list as follows:
The following theorem has been brought to my attention:
Take two non-degenerate conics in the projective plane. There are four lines which are tangent to both conics, giving eight contact points in total.
There exists another conic, which runs through these eight points.
(a) How well known is this theorem?
(b) Does it have a name, and if so, what?
(c) Where can I find a proof?
@BernhardWerner First reply:
Dunno offhand --- but I'd bet it's a special case of something more general, and probably already well-known.
Googling << Bitangents to planar quartic curve >> turns up a heap of stuff, going back to Pluecker in the XIX-th century, but much of it surprisingly recent. It might be worthwhile to make contact with some of those authors.
A more general starting point:
@BernhardWerner I know you have a proof for the conic version you mentioned, and that the proof seems to be novel and nice, but it's worth knowing what already exists.
@ColinTheMathmo Oh, definitely! Currently, it's floating around in empty space. Would be much better if it relates to something.
@BernhardWerner Another reply, and as is common, they aren't actually answering the question:
I wonder if this is easier to see or to prove if you look at it in the dual space.
In the space of lines in the plane, the set of lines tangent to a given conic is also a conic, the dual conic.
Given two conics, their duals intersect in 4 points, because two quadratics intersect in 4 points.
The tangents to the two conics at these 4 points give 8 lines.
The theorem then says that there is another conic tangent to these 8 lines.
I have replied to thank them, but also to point out that they have answered a question I didn't ask, and didn't answer any of the questions I did ask.
@ColinTheMathmo Thank you! Yes, I also considered the dual case. But any (algebraic) proof will most likely smash some matrices and their inverses together, which can then be interpreted either way around. I chose the one which appeared nicer/simpler in my mind.
But, yes, the primal and dual statements are equivalent.
And thanks again for sending information back and forth!
@BernhardWerner The trend continues, people engage with the problem, but don't answer the questions asked:
Here's a geogebra demo of two conics and their tangents creating a third.
This isn't a proof, but geogebra certainly seems to believe it.
You can move the first ten points (A-J) around to change the "source" conics, and watch the tangent lines, the eight points from those tangencies, and the resulting third conic change.
@BernhardWerner (cont'd) ...
This is plane geometry, not projective geometry, so sometimes you don't get tangent lines if you move the points around poorly.
So we still haven't got an answer to the question "How well known is this?"
I've posted a follow-up pointing out that while I love the engagement, no one has answered the question. We'll see if it provokes a response.
(I'm beginning to doubt it)
@ColinTheMathmo Thank you so much, again, for doing all this! It seems my little theorem is at least a bit interesting But let's hope for some actual answers, yes!
@ColinTheMathmo Ha, that's great! I have my version of this wirtten in CindyJS. (A bit numerically unstable, unfortunately.)
I kinda feels that theorem looks like have been proved by someone before, I mean intuitively we can see that there must be 4 tangent lines tangent to both of them (at least I can imagine that for circle or ellipse) and someone most likely have thougth about it and try to prove it.
@swordofgod Yes, the starting situation looks too obvious to not have been analysed already.
Maybe it was lost in the replies, but I did, in fact, find a reference to this in a book from 1913. Hatton shows that it's equivalent to a characterization via cross-ratios:
But that's most likely not the earliest someone has proven it.
@BernhardWerner I will certainly do that for you. I will use it as an opportunity to mention mathstodon as well.
It's a fun thing you found.
Is your proof still new/novel/interesting? Or is it subsumed by the material in the book?
@ColinTheMathmo My proof is still interesting (I hope) and comes at it from an entirely different angle. Will link to the paper once it's up on the arXiv, but:
I was looking at a certain tensor diagram (the way Jim Blinn uses them) for which I haven't found a geometric interpretation anywhere.
After a while, I stumbled over the theorem I posted here and wondered what it meant.
So, what I did revolves around an algebraic oddity for which this theorem is the geometric interpretation.
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