@Akiva I assume you consider, like, a shape like "V" a polygon and what not

You can easily limit to that if you take intersection of nonconvex chaps

Decreasing intersection of polygons of exactly $n$ sides should in general be a polygon of $n$ sides, with degeneracies (which are polygons with certain vertices and edges squished together)

so you also get shapes like polygons with an edge sticking out etc etc etc

I see @Balarka is still not un-sleeping.

@ALannister: so write $3(a+b\sqrt{-5}) = 2+\sqrt{-5}$ and you get $3a=2$ and $3b=1$, with $a,b\in\Bbb Z$. Huh?

Trying to solve a number theory problem.

@TedShifrin by any chance, do you know anything about subderivatives?

No, not really.

oh ok

:)

btw, what kind of shape is your logo a 3D model of anyway?

What do you think?

it looks like something mathematical, but I don't recognize it.

probably a parametric surface of some form

Hint: It's the shape you get when you spin some common object around an axis (the axis being drawn in the picture).

oh it's just a surface of revolution?

but then why does the middle look twisted? Is that just the way the 3D engine drew it?

@ted do you know a good source to start learning variational calculus

Did that kind of stuff come up in your work?

It's not twisted ... It's just showing the actual things spinning around.

oooh, lol.

@PVAL: No, not really. At what level are you trying to see it?

I thought maybe it was some kind of complicated tube plot with a variable rotation and radius as it moves along the curve

google.com/…

I'd like to understand some of the classical theorems of Poincare on geodesics of spheres and that nature.

It is a good linear algebra exercise, GreatDuck. Plus actually understanding what's going on.

@TedShifrin The tube plot or the surface?

Geodesics of spheres? What theorems of Poincaré?

I don't know what you mean by tube plot, GreatDuck.

@TedShifrin images of tube plots google.com/…

My surface is not a tube around a curve.

ah.

A cone is not a tube, for starters ...

e.g. any surface of positive curvature has a closed geodesic is probably the simplest example I know of.

some have variable radii and rotate as they go along the curve

Although tubes make for a really good undergraduate differential geometry exercise.

Oh, I don't know a calculus of variations proof for that, @PVAL.

What's the proof you know?

yeah, I made a tube plot thing in the computer graphics course last semester. making code to replicate the derivative was definitely intriguing

^one with variable radius

My understanding was that sort of falls out looking for critical points of the length functional for the space of geodesics on the surface.

Anyhow, Gelfand-Fomin and Bliss are standard good texts, I think (old fashioned, though).

I dunno how you see it's closed that way, @PVAL. I guess I thought I remembered some sort of infinite-dimensional Morse theory argument, but I haven't thought about this in centuries.

My main goal is to understand Arnold's 50 page book on immersions of closed curves into planes, and that uses similar ideas.

Ah.

@Ted Isn't what I said some sort of morse theory argument?

Griffiths wrote a cute little book on calculus of variations and geometric problems using moving frames.

@TedShifrin probably could replicate that with a tube-plot-like parameterization using the type I'm referring to but I see what you mean about it not really being the right way to create it or the most straightforward.

But calculus of variations would usually be applied to paths with fixed endpoints, @PVAL. I suppose you can restrict yourself to maps of a circle.

Well, any surface of revolution is a generalized tube in your sense, GreatDuck, but I don't find this productive.

Do you know anything about the arnold book I am talking about?

Nope.

@TedShifrin true. I wasn't so much referring to the mathematical usage though as I was referring to the means of creating the 3D model, in which case it might be easier to just make a tube plot than to make a tube plot and a surface of revolution. More code = more bugs. We don't want roaches.

btw, that comment you made yesterday still cracks me up.

I have no idea what you're talking about ...

somebody was being a bit impatient it would seem.

Oh ... the fast-food comment.

yup

He defines 3 invariants of generically immersed plane curves which are invariant under regular homotopies of immersions with no triple points and self-tangencies, by defining them for the standard turning number immersion with the least number of self-intersections and defining the jumps over triple intersections and self-tangencies.

glad to see people like you putting the more impatient people in their place. :-)

Danu and Demonark called the person out, not I ... I merely sat on the sidelines and made that comment.

well, a snarky comment like that sometimes does more than a scolding tongue in my experience.

I think the proof that these are invariants is mainly contact geometry (and doesn't require calculus of variations).

anyway, I'm probably going to head out.

Yeah, sounds more like singularity theory ideas than calculus of variations, @PVAL.

Bye, GreatDuck.

Oh

Idk what its called.

But my understanding is its the same kind of ideas as in the theorem of Poincare I quoted.

Arnol'd did a lot of stuff with singularities of maps. (McCrory and I wrote a JDG paper, as it turned out, doing all the technical details that Arnol'd skipped over. And making the Chern class arguments clearer, too.)

looking at some hypersurface inside some infinite dimensional space and seeing what happens when you cross the hypersurface.

That sounds more like singularity theory to me. I dunno. I'd have to read the book.

I've looked at his big book on isolated singularities I think.

but the thing I am reading now is much more relevant to me and much more accessible.

@Ted This is the book bookstore.ams.org/ulect-5

Yeah, looks like Legendrian singularities to me. ... I mean, they mention Jacobi fields and conjugate points, but that's only one little thing.

But hell if I know.