On Wednesday 16 April 2003 08:36 pm, you wrote:
Suppose one bank is a perfectly straight line and the other is infinitely
ragged. The midline would be half as ragged as the infinitely ragged bank,
i.e. infinitely ragged.
BJB

> hmmm - I'm not a math expert by any means, but I
> thought there would be some sort of convergence
> here...the "midline" has to lie somewhere, and it is
> possible for there to be a finite and accurate midline
> between two infinite lines. Given that the midline
> would be a function over a finite distance, I think
> the length of the midline would be finite also.
>
> --- "Brian J. Butler" <bjbutler@...>
>
> wrote:
> > On Wednesday 16 April 2003 07:12 pm, you wrote:
> > The thalweg is also of indefinite length because
> > there can be an infinite
> > number of changes in the y value over any interval
> > of x.
> > BJB
> >
> > > The riverbanks can have an almost infinite length
> > > which is practically impossible to measure, but I
> > > think the center or thalweg can not be considered
> >
> > in
> >
> > > the same category. Looking at it from above with
> >
> > one
> >
> > > bank on the bottom and the other on top (thus
> > > traveling along the bank is traveling along the
> >
> > x-axis
> >
> > > and crossing river is along y-axis), there emerges
> >
> > an
> >
> > > upper and lower bound. When drawing the boundary
> > > along the center or thalweg, you can only go in
> >
> > one
> >
> > > direction without doubling back and must stay
> >
> > within
> >
> > > the bounds...essentially only one y-coordinate per
> > > each x-coord. This is unlike the bank which, due
> >
> > to
> >
> > > shapes of rocks, you can have the shore go in and
> >
> > out
> >
> > > (several y-coord per x-coord).
> > >
> > > --- "Brian J. Butler" <bjbutler@...>
> > >
> > > wrote:
> > > > On Wednesday 16 April 2003 06:20 pm, you wrote:
> > > > Why would the center of a river be any smoother
> >
> > than
> >
> > > > banks from which it is
> > > > equidistant? And molecules, grains of sand,
> > > > boulders, and great bends are all
> > > > features that determine the edge of a river,
> >
> > hence
> >
> > > > its length. For some
> > > > discussions it might be OK to ignore features
> >
> > below
> >
> > > > a certain size, but we
> > > > are specifically discussing the length of the
> > > > boundary, which is dependent on
> > > > the scale of objects we consider.
> > > > BJB
> > > >
> > > > > The center of a river is about as smooth as
> >
> > you
> >
> > > > can get.
> > > >
> > > > > Why would you measure a boundary line around a
> > > >
> > > > grain of sand?
> > > >
> > > > > Theoretically, this discussion could get into
> >
> > such
> >
> > > > things, but practically,
> > > >
> > > > > the line would go right across the top of that
> > > >
> > > > grain, or boulder, or rock,
> > > >
> > > > > not around it.
> > > > >
> > > > > -----Original Message-----
> > > > > From: Brian J. Butler
> > > >
> > > > [mailto:bjbutler@...]
> > > >
> > > > > Sent: Wednesday, April 16, 2003 3:44 PM
> > > > > To: BoundaryPoint@yahoogroups.com
> > > > > Subject: Re: [BoundaryPoint] Re: American
> >
> > State
> >
> > > > Boundaries
> > > >
> > > > > On Wednesday 16 April 2003 05:12 pm, you
> >
> > wrote:
> > > > > You are not on the right wavelength yet. The
> > > >
> > > > natural boundaries you
> > > >
> > > > > enumerated are not smooth curves that can be
> > > >
> > > > measured in the traditional
> > > >
> > > > > sense. I agree that you can determine a
> >
> > minimum
> >
> > > > length of these boundaries
> > > >
> > > > > by interpolating between fixed points on the
> > > >
> > > > boundary. But the true length
> > > >
> > > > > of the boundary depends on how small your
> >
> > samples
> >
> > > > are. For example, you
> > > >
> > > > > would have a longer measurement if you
> >
> > measured
> >
> > > > around each rock along the
> > > >
> > > > > riverbank, or each grain of sand. So you are
> > > >
> > > > doubly correct - your
> > > >
> > > > > estimate
> > > > >
> > > > > could be off by a great margin, an infinite
> >
> > margin
> >
> > > > perhaps, and the minimum
> > > >
> > > > > length of the OK-TX boundary is longer than
> >
> > the
> >
> > > > CA-NV boundary. I don't
> > > >
> > > > > think you can make the statement that the
> >
> > OK-TX
> >
> > > > boundary is longer than the
> > > >
> > > > > VA-WV boundary, though, for example, because
> >
> > it
> >
> > > > depends on how irregular
> > > >
> > > > > the
> > > > >
> > > > > boundaries are and how carefully you measure
> >
> > them.
> >
> > > > > BJB
> > > > >
> > > > > > Well, *anything* has a length depending on
> >
> > how
> >
> > > > you measure it. But most
> > > >
> > > > > > US state boundaries have specific
> >
> > definitions
> >
> > > > that are actual places on
> > > >
> > > > > > the ground, whether it's mean highwater,
> >
> > center
> >
> > > > of channel. top of the
> > > >
> > > > > > ridge, etc. E.g., the Kentucky boundary
> >
> > along
> >
> > > > the Ohio River is the
> > > >
> > > > > > waterline on the northern bank, so KY
> >
> > controls
> >
> > > > the river. The definition
> > > >
> > > > > > IIRC was fixed in time so that it doesn't
> >
> > change
> >
> > > > as the Ohio rises or
> > > >
> > > > > > falls or carves new segments of the channel.
> > > > > >
> > > > > > Anyway, I am still curious if there is a
> >
> > place
> >
> > > > to ascertain the actual
> > > >
> > > > > > length of the OK-TX boundary? I estikated it
> >
> > as
> >
> > > > somewhere around 800
> > > >
> > > > > miles,
> > > > >
> > > > > > while CA-NV was just over 600 miles. My
> > > >
> > > > estimates could be off by a great
> > > >
> > > > > > margin, but I don't think they are off so
> >
> > much
> >
> > > > as to change the ranking.
> > > >
> > > > > > -----Original Message-----
> > > > > > From: Brian J. Butler
> > > >
> > > > [mailto:bjbutler@...]
> > > >
> > > > > > Sent: Wednesday, April 16, 2003 12:12 PM
> > > > > > To: BoundaryPoint@yahoogroups.com
> > > > > > Subject: Re: [BoundaryPoint] Re: American
> >
> > State
> >
> > > > Boundaries
> > > >
> > > > > > On Wednesday 16 April 2003 01:18 pm, you
> >
> > wrote:
> > > > > > A natural boundary, such as a river, has a
> > > >
> > > > length that depends on how
> > > >
> > > > > > closely
> > > > > > you measure it.
> > > > > > BJB
> > > > > >
> > > > > > > But CA-NV wouldn't be the longest border
> > > >
> > > > between two states, straight
> > > >
> > > > > > > or not. The OK-TX border for a good
> >
> > distance
> >
> > > > is the meandering Red
> > > >
> > > > > > > River. There's no basis to say that
> >
> > doesn't
> >
> > > > count as distance and that
> > > >
> > > > > > > one should draw an imaginary "straight"
> >
> > line
> >
> > > > instead to cut the
> > > >
> > > > > > > corners. The boundary is the boundary line
> > > >
> > > > itself.
> > > >
> > > > > > > What is the length of the CA-NV boundary
> >
> > and
> >
> > > > the OK-TX boundary?
> > > >
> > > > > > > -----Original Message-----
> > > > > > > From: Brian J. Butler
> > > >
> > > > [mailto:bjbutler@...]
> > > >
> > > > > > > Sent: Wednesday, April 16, 2003 9:39 AM
> > > > > > > To: BoundaryPoint@yahoogroups.com
> > > > > > > Subject: Re: [BoundaryPoint] Re: American
> > > >
> > > > State Boundaries
> > > >
> > > > > > > On Wednesday 16 April 2003 11:19 am, you
> > > >
> > > > wrote:
> > > > > > > CANV is certainly the straight-line champ.
> >
> > If
> >
> > > > we "go fractal" maybe
> > > >
> > > > > > > ID-MT, OK-TX, or even VA-WV would take the
> > > >
> > > > cake.
> > > >
> > > > > > > BJB
> > > > > > >
> > > > > > > > nice question & nice answer
> > > > > > > >
> > > > > > > > how about canv for longest
> > > > > > > >
> > > > > > > > --- In BoundaryPoint@yahoogroups.com,
> >
> > Brian
> >
> > > > J. Butler
> > > >
> > > > > > > > <bjbutler@b...> wrote:
> > > > > > > > > On Wednesday 16 April 2003 09:50 am,
> >
> > you
> >
> > > > wrote:
> > > > > > > > > The shortest is easy - at AZ-CO-NM-UT
> > > >
> > > > there are two pairs of
> > > >
> > > > > > > > states that meet
> > > > > > > >
> > > > > > > > > at a point.
> > > > > > > > > BJB
> > > > > > > > >
> > > > > > > > > > Which state shares the longest
> >
> > border
> >
> > > > with another state?
> > > >
> > > > > > > > (The border
> > > > > > > >
> > > > > > > > > > does not have to be continuous.)
> >
> > Which
> >
> > > > state shares the
> > > >
> > > > > > > > shortest?
> > > > > > > >
> > > > > > > > > > Your use of Yahoo! Groups is subject
> >
> > to
> >
> > > > > > > > http://docs.yahoo.com/info/terms/
> > > > > > > >
> > > > > > > > > --
> > > > > > > > > Brian J. Butler
> > > > > > > > > BJB Software, Inc.
> > > > > > > > > 508-429-1441
> > > > > > > > > bjbutler@b...
> > > > > > > > > http://www.bjbsoftware.com
> > > > > > > >
> > > > > > > > Your use of Yahoo! Groups is subject to
> > > > > >
> > > > > > http://docs.yahoo.com/info/terms/
> > > >
> > > > --
> > > > Brian J. Butler
> > > > BJB Software, Inc.
> > > > 508-429-1441
> > > > bjbutler@...
> > > > http://www.bjbsoftware.com
> > >
> > > __________________________________________________
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> > >
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> >
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> >
> > --
> > Brian J. Butler
> > BJB Software, Inc.
> > 508-429-1441
> > bjbutler@...
> > http://www.bjbsoftware.com
>
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--
Brian J. Butler
BJB Software, Inc.
508-429-1441
bjbutler@...
http://www.bjbsoftware.com