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. 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... Then later in the same paper "Definition 2. A polygon is a Jordan curve that is a subset of a finite union of lines. A polygonal path is a continuous function P : [0, 1] ->¨ R2 that is a subset of a finite union of lines. It is a polygonal arc, if it is 1 . 1." By that definition a polygonal path is not a curve. Worse: A polygonal path is a function from [0,1] to R^2 _that is a subset of a finite union of lines_. There's no such thing - the _image_ of such a function can be a subset of a finite union of lines. Not that it matters, but his defintion of "polygonal path" is, _if_ we're being very careful, self-contradictory. So I don't think we can count that paper as a suitable reference for what the _standard_ definitions are; the standard definitions are not self-contradictory this way. Then the second definition you cite: Amazon says the prerequisites are two years of calculus. The stanard meaning of log is log base e, even though it means log base 10 in calculus. >Mark -- http://mail.python.org/mailman/listinfo/python-list