On Jun 22, 2009, at 2:16 PM, David C. Ullrich wrote:
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.)
That is to say, the "winding number" is a property of both the curve
and a parameterization of it. Or, in other words, the winding number
is a property of a function from S1 to C.
Charles Yeomans
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