On Mon, 22 Jun 2009 10:31:26 -0400, Charles Yeomans <char...@declaresub.com> wrote:
> >On Jun 22, 2009, at 8:46 AM, pdpi 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: >>> >>> >>> <snick> >>> >>> >>> >>>> 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 ->ยจ 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. >> -- > > >But certainly the image set does define the curve, if you want to view >it that way -- all parameterizations of a curve should satisfy the >same equation f(x, y) = 0. This sounds like you didn't read his post, or totally missed the point. Say S is the set of (x,y) in the plane such that x^2 + y^2 = 1. What's the "index", or "winding number", of that curve about the origin? (Hint: The curve c defined by c(t) = (cos(t), sin(t)) for 0 <= t <= 2pi has index 1 about the origin. The curve d(t) = (cos(-t), sin(-t)) (0 <= t <= 2pi) has index -1. The curve (cos(2t), sin(2t)) (same t) has index 2.) >Charles Yeomans -- http://mail.python.org/mailman/listinfo/python-list