On Mon, 22 Jun 2009 05:46:55 -0700 (PDT), pdpi <pdpinhe...@gmail.com> wrote:
>On Jun 19, 8:13 pm, Charles Yeomans <char...@declaresub.com> wrote: >> On Jun 19, 2009, at 2:43 PM, David C. Ullrich wrote: >> >> >> >> >> >> > Evidently my posts are appearing, since I see replies. >> > I guess the question of why I don't see the posts themselves >> > \is ot here... >> >> > On Thu, 18 Jun 2009 17:01:12 -0700 (PDT), Mark Dickinson >> > <dicki...@gmail.com> wrote: >> >> >> On Jun 18, 7:26 pm, David C. Ullrich <ullr...@math.okstate.edu> >> >> wrote: >> >>> On Wed, 17 Jun 2009 08:18:52 -0700 (PDT), Mark Dickinson >> >>>> Right. Or rather, you treat it as the image of such a function, >> >>>> if you're being careful to distinguish the curve (a subset >> >>>> of R^2) from its parametrization (a continuous function >> >>>> R -> R**2). It's the parametrization that's uniformly >> >>>> continuous, not the curve, >> >> >>> Again, it doesn't really matter, but since you use the phrase >> >>> "if you're being careful": In fact what you say is exactly >> >>> backwards - if you're being careful that subset of the plane >> >>> is _not_ a curve (it's sometimes called the "trace" of the curve". >> >> >> Darn. So I've been getting it wrong all this time. Oh well, >> >> at least I'm not alone: >> >> >> "De?nition 1. A simple closed curve J, also called a >> >> Jordan curve, is the image of a continuous one-to-one >> >> function from R/Z to R2. [...]" >> >> >> - Tom Hales, in 'Jordan's Proof of the Jordan Curve Theorem'. >> >> >> "We say that Gamma is a curve if it is the image in >> >> the plane or in space of an interval [a, b] of real >> >> numbers of a continuous function gamma." >> >> >> - Claude Tricot, 'Curves and Fractal Dimension' (Springer, 1995). >> >> >> Perhaps your definition of curve isn't as universal or >> >> 'official' as you seem to think it is? >> >> > Perhaps not. I'm very surprised to see those definitions; I've >> > been a mathematician for 25 years and I've never seen a >> > curve defined a subset of the plane. >> >> I have. >> >> >> >> >> >> >> >> > Hmm. You left out a bit in the first definition you cite: >> >> > "A simple closed curve J, also called a Jordan curve, is the image >> > of a continuous one-to-one function from R/Z to R2. We assume that >> > each curve >> > comes with a fixed parametrization phi_J : R/Z ->¨ J. We call t in R/Z >> > the time >> > parameter. By abuse of notation, we write J(t) in R2 instead of phi_j >> > (t), using the >> > same notation for the function phi_J and its image J." >> >> > Close to sounding like he can't decide whether J is a set or a >> > function... >> >> On the contrary, I find this definition to be written with some care. > >I find the usage of image slightly ambiguous (as it suggests the image >set defines the curve), but that's my only qualm with it as well. > >Thinking pragmatically, you can't have non-simple curves unless you >use multisets, and you also completely lose the notion of curve >orientation and even continuity without making it a poset. At this >point in time, parsimony says that you want to ditch your multiposet >thingie (and God knows what else you want to tack in there to preserve >other interesting curve properties) and really just want to define the >curve as a freaking function and be done with it. Precisely. -- http://mail.python.org/mailman/listinfo/python-list