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.
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."
These are a bit too casual for me as well...
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.
Charles Yeomans
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