Theorem of Curves in higher dimensions

[eddo]

I just finished learning the fundamental theorem of curves in 3 dimensions. As a reminder, this is the theorem that states that a continuous, C infinity, unit speed curve in 3d is uniquely determined by its curvature and torsion (up to actions by SE(3), that is rotations and translations).

My question is, how does this generalize to higher dimensions? I suspect that in n dimensions, you would need n-1 Real functions ( in 3d we have the curvature and torsion) to uniquely determine a unit curve up to actions under SE(n). My reasoning is that curvature in a sense tells the curve how to move in a plane, and torsion tells it how to move off the plane, so for each extra dimension you would need one additional number, to tell it how to move in the extra "direction".

Does anyone know if this is right, and how you could prove it? My prof had never heard of a generalization of this sort. Could anyone direct me to some books, journals etc... which deal with this? Thanks.

 

[eddo]

Alas, my good friend Wikipedia has answered my question for me. Here's the link for those who are interested:

http://en.wikipedia.org/wiki/Curvature_vector

It generalizes to n-dimensions pretty much how you would expect. I found the part about how the curvature vectors are simply found by Gram-Schmidt, and that the generalized curvatures are based on this in a rather simple way very interesting. Seeing the general case makes the curvature and torsion seem a lot less mysterious. I recommend anyone who hasn't seen this take a look at the link. The generalization of the Frenet-Serret formulas is very nice and simple as well. Enjoy.

 

[robphy]

You may be interested in http://arxiv.org/abs/gr-qc/0601002
On the differential geometry of curves in Minkowski space
J. B. Formiga, C. Romero

We discuss some aspects of the differential geometry of curves in Minkowski space. We establish the Serret-Frenet equations in Minkowski space and use them to give a very simple proof of the fundamental theorem of curves in Minkowski space. We also state and prove two other theorems which represent Minkowskian versions of a very known theorem of the differential geometry of curves in tridimensional Euclidean space. We discuss the general solution for torsionless paths in Minkowki space. We then apply the four-dimensional Serret-Frenet equations to describe the motion of a charged test particle in a constant and uniform electromagnetic field and show how the curvature and the torsions of the four-dimensional path of the particle contain information on the electromagnetic field acting on the particle.

 

[eddo]

That sounds very interesting. I'll have to take a good look at that when I have time.

I actually noticed some very serious mistakes in the statement of the theorem on Wikipedia. First of all a minor type: they keep saying deviance when they mean deviation. Much more importantly, in the theorem they talk about n Frenet vectors e_n, and n chi functions, but by the definition of the chi functions, there can only be n-1 chi functions.

Also, I believe this off by 1 error is carried over into the next line as well, where they say that the first n-1 chi vectors are strictly positive, it should state the the first n-2 chi vectors are strictly positive. For example, in the 3 dimensional case, the curvature is strictly positive, and the torsion takes on any value. If the torsion were required to be strictly positive, than a lot of curves would be missed by the theorem, since a curve with positive curvature and negative torsion cannot be gotten from one with positive curvature and positive torsion by transformations in SE(3) (because it would require a reflection).

It seems to me that having the last chi be able to take on positive and negative values allows this to take the place of any reflection (simply by rotating and translating appropriately) so that the theorem captures all smooth curves. Can anyone who knows more verify this?

 

[Doodle Bob]

The go-to guys for Frenet frames in higher dimensions are V.I.Arnold and R. Uribe-Vargas. See http://arxiv.org/pdf/math.DG/0504132 for a brief look at the current state of affairs.

You are correct about the problems with the Wikipedia entry. Your assertion at the end should work for all smooth curves that are always dimensionally full, i.e. have no points with a zero curvature (of any dimension) anywhere. It is possible after all for a curve in 3-space to have 0 curvature at exactly one point.

Torsion of course can have positive, negation and *zero* values anywhere. In fact, the n-dimensional vertex theorems state that a simple, closed curve in R^n has at least so-many (I forget now, something like n+2) vertices.

 

[ObsessiveMathsFreak]

For a bite sized summary of the Arnol'd Vargas paper, try this, section 1.12.

As a curve moves about, it locally lies "within" lines, and planes. That is to say, if you zoom in enough, the curve "almost" fits on a paticular line, and "almost" fits into a 2d plane.

Curvature ($$\kappa_1$$)is a measure of how fast the curve is moving out of the current line it "lies" in.
Torsion ($$\kappa_2$$ or $$\tau$$)is a measure of how fast the curve is moving out of the current plane it "lies" in.

The next higher dimensional curvature would be $$\kappa_3$$, a measure of how fast the curve is moving out of the 3d volume it finds itself in.
$$\kappa_4$$ how fast its moving out of its current 4d frame, etc.

 

[Chris Hillman]

Wikipedia? Watch out!

I'd add to what eddo wrote that ALL students should always be VERY cautious in reading ANY Wikipedia article. As a former active contributor to the math/science articles (more than 10,000 edits from May 2005 through August 2006; see http://en.wikipedia.org/wiki/User:Hillman/Archive for some indication of what I initially hoped to accomplish at Wikipedia--- I can't claim that the versions listed there are entirely error-free, just that I reviewed them closely and tried hard to keep errors out, to the degree this was consistent with respecting the views of other WP editors), I know all too well that:

1. some editors deliberately introduce errors, including hard to spot ones like sign errors (yes, I have specifically seen editors going around math articles doing precisely that), apparently because they enjoy antisocial activities,

2. some editors are very ill-informed or extremely bad writers, but don't hesitate to make large edits which largely destroy the value of previously good articles (one very common problem arises when an inexperienced writer inserts some new material without noticing that he has destroyed the flow of ideas or mangled a previously sensical segue into the next paragraph in the immediately preceding version--- I refer to phenomena like this which tend to gradually degrade articles over time, until a more knowledgeable and experienced editor takes the time to extensively rewrite and reorganize a long but disorganized and unreadable article, as "edit creep"),

3. some editors are on a mission to misleadingly portray (often as "anons" or under a "false flag") their own (obscure, inchoate, controversial or even downright cranky) "theory" as representing mainstream scientific opinion; while phenomena like "wikishilling" are often fairly easy to spot if you are experienced enough to be looking out for it, in some cases advanced knowledge of the subject in question might be needed to immediately tell when this is happening.

Wikipedia articles are also highly unstable because anyone can edit any article (well almost anyone can edit almost any article) at any time. If you feel you simply MUST cite a Wikipedia article, you should always use the "Permanent link" button (look at the left sidebar when you visit WP) to obtain a url pointing to the specific version which you read. But you should always consult printed sources and make a sustained and serious attempt to verify that what some version of a WP article claims seems consistent with what you read at independent sources. (Note that many "competing" on-line encyclopedias copy WP articles more or less verbatim en masse, as they are permitted to do by the GPL and related licenses, but it might not always be clear when this has happened because WP articles often change rapidly and drastically, particularly when they concern topics which have recently been in the news.)

On the other hand, some math related articles have been quite good and largely error-free for extended periods of time. Many experienced Wikipedians have observed that many "obscure" mathematical topics are well covered if one of the WikiProject Mathematics members happens to be a devoted fan, but unfortunately, one loon or troll can cause an utterly disproportionate amount of damage in a short time.

I feel there is a pressing need for a stable website offering advice to students, teachers, journalists, jurists, lawmakers, and others who extensively use Wikipedia as an information resource (scary? yes, scary!), since it seems clear that it has come to be generally regarded as too valuable not to use simply because it lacks any real concept of information integrity. I see grave social implications in the increasingly prevalent notion that "good enough information" means "plausible-appearing" (to a non-expert), rather than "authoritative", much less "true".

I actually attempted to write in my Wikipedia "user space" several essays describing my views on the promise and dangers posed by the Wikipedia for the citizenry of Earth, but Wikipedia has no provision for signed essays which cannot be vandalized by disgruntled editors holding a contrary view, and eventually I felt compelled to abandon my attempts to provide in Wikipedia itself an honest but searching examination of the challenges facing the Wikipedia in the so-called "year of quality" announced earlier in 2006 by cofounder Jimmy Wales. At some point I might take up this project again in another venue, however.

Chris Hillman

 

[Doodle Bob]

3. some editors are on a mission to misleadingly portray (often as "anons" or under a "false flag") their own (obscure, inchoate, controversial or even downright cranky) "theory" as representing mainstream scientific opinion; while phenomena like "wikishilling" are often fairly easy to spot if you are experienced enough to be looking out for it, in some cases advanced knowledge of the subject in question might be needed to immediately tell when this is happening.


Chris Hillman

I always thought this might be true. And I have noticed a particular example of this recently: The sections on 4-dimensional polytopes mentions that these are often called "polychorons." I was taken aback since I had never heard of the term. Certainly, the experts of the polytope field that I knew of (e.g. Grunbaum, Ziegler, Readdy) never used the term. So, I went immediately to MathSciNet (AMS's massive mathematical article database) and searched for anything containing the term: not a single article. Going back to the Wiki article, I found that only one person in the world, as far as I know, actually uses that term: namely, the guy whose website is referenced on the Wiki article and presumably the guy who wrote the Wiki article.

 

[Chris Hillman]

Polychorons

I always thought this might be true. And I have noticed a particular example of this recently: The sections on 4-dimensional polytopes mentions that these are often called "polychorons." I was taken aback since I had never heard of the term.

I agree, this term is nonstandard. BTW, Grunbaum served on my thesis committee! And Lou Billera was on the faculty at my undergraduate school. Not surprisingly, I have a long standing interest in polytopes and enumerative combinatorics a la Stanley.

Going back to the Wiki article, I found that only one person in the world, as far as I know, actually uses that term: namely, the guy whose website is referenced on the Wiki article and presumably the guy who wrote the Wiki article.

Yes, while I wasn't thinking of that particular article, I actually was aware of it before I left WP, but those of us who were trying to correct this kind of thing were overwhelmed by problems of this type. By the time I left, there were only a handful of users making much attempt to fix this kind of thing, and no-one was trying to do so on a systematic basis. If you want to try to inject some new life into efforts to try to reign in rampant misinformation about science, you might stop by http://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Pseudoscience, which at one time had a few dozen members who tried to monitor and fix bad science articles (not neccessarily limited to outright crank articles). It looks like there has been almost no activity there since I left, unfortunately.

The biggest problem is that WP policies and administrative procedures for dealing with bad articles (hoax articles, crank articles, wikshilling, etc.) haven't scaled gracefully and have been woefully inadequate for quite some time. The biggest problem in reforming these is, in my view, lack of any real leadership from Jimbo Wales (despite his proclamation of the "Year of Quality") and consequently, failure to reform even the prodecure for reforming outdated and inadequate policies and procedures, which is the reason why attempts to implement stronger policies intended to improve the integrity of information in the Wikipedia have to date all failed.

Chris Hillman