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From: Archimedes Plutonium on 7 Feb 2010 16:57


gudi wrote:
> Hi Archimedes Plutonium,
>
> Without differential geometry basics I am afraid you cannot progress
> much in
> obtaining a satisfying answers for your questions.
>
> Just as surfaces in flat Euclidean geometry are developable and can be
> made to
> contact everywhere, in doubly curved geometry we have the
> corresponding concept
> that is called " applicability". However there is only local isometry,
> not global
> isometry. It is referred to as being " in the small", or in local
> neighborhoods,
> and in contrast we have " in the large" etc. We have to be content
> with contact
> along lines at a single point only.
>
> " Geometry and Imagination " is a good book. Many Catenoid to Helicoid
> bendings Java
> simulations are available on the net.They appear to be two different
> shapes but can be bent
> from one shape to the other shape i.e., by deforming it. You can also
> practically take a thin
> plastic water container with narrow waist part cut out and get a helix
> by a single cut along
> meridian and then stretching out.
>
> During an "application" between helicoid/catenoid thin foils, Gauss
> curvature remains the same,
> at each corresponding point during the deformation.
>
> As to your question of where exactly the contact is established, in
> surface theory, we have
> for each point( x,y,z ) = ((f(u,v),g(u,v)) parameterization where the
> parameters u and v describe the
> grid lines making up the curved surface in a curvilinear coordinate
> system. The choice
> is entirely yours. You can choose an orthogonal system, geodesic polar
> coordinates, asymptotic
> lines of a Chebycheff Net etc. and go along any u- line or v- line of
> your choice.
>
> A special case of Gauss Egregium theorem is the Minding therem that
> states that for
> surfaces class >=3,in the small or locally, an isometry is possible
> between two surfaces of
> constant equal Gauss curvature.
>
> The earlier sketch is modified somewhat here:
>
> http://i46.tinypic.com/f0rm9u.jpg
>
> However, it serves no purpose in understanding of "applicability" of
> surfaces. In isometry the
> area remains same ( through E*G - F^2 determinant of first
> fundamental form) but in the converse,
> if your question is, the area is same and what straining of u- and v-
> lines is to be chosen to
> get back isometry of each du and dv element, one has to use Beltrami
> differential operators etc.,
> I think it takes lot more effort.
>
> Narasimham

Narasimham, can you please do another sketch of just the grid block of
your graph
of R from 1.0 - 0.8 and then of Z from 0.0 - 0.2. That is, just expand
and make as
big as possible that graph block showing what happens with the
pseudosphere
curve and sphere curve inside that grid block?

I mentioned several courses of proof that there exists or does not
exist a "arc length"
that is identical on both curves. An obvious proof route would be the
analytical route
where we obtain the equation for the path of the line on the
pseudosphere (tractrix) and
the equation for the path of a great-circle (both objects having equal
radii). The Analytic
proof would show that the coordinate point exists on both, once a
movement or translation
of the arc is made to happen.

So I think, that if there exists such a small arc that is replaceable
on both the tractrix with
the sphere great-circle that this arc would lie in the beginning of
the curve from about
0 to before it becomes 1. In other words, where this equal and
replaceable arc lies is in
the interval 0 to 1 where the numbers are fractions. And, as we know,
in Hyperbolic geometry
it starts out appearing "normal as Euclidean" and then as we go
further and further along a "line" we are far from normal. So what I
think, is that there is a small arc of say, about 10% of
the arc length of an entire great-circle which is replaceable by a arc
of the tractrix.

Narasimham talks about a one-point flexibility of bending a tractrix
arc into coinciding with a
great-circle arc. The exercise may come down to a flex-bending, but I
suspect that it may
be unnecessary. I suspect that no-one in the history of mathematics
ever had a notion to
look around and pick through the tractrix looking for such an equal
arc. And I myself, would
have never had the nerve to search through, if not for this idea that
Algebra fizzles out at
about 10% of all the numbers and where there is no trustworthy algebra
for 90% of all the numbers that exist. The 10% is what I call the
Incognitum between Finite and Infinity and this
10% is showing up also in the Luminet-Poincare Dodecahedral Space.

Please, Narasimham, can you do another sketch of just that grid block
expanded to as large
as you can make it.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
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