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From: Archimedes Plutonium on 6 Feb 2010 22:51


gudi wrote:
> On Feb 7, 2:33 am, Archimedes Plutonium
> <plutonium.archime...(a)gmail.com> wrote:
> > Archimedes Plutonium wrote:
> > > gudi wrote:
> > > > I don't understand what you understand by "Rolling".  Without looking
> > > > into any other details in your postings and
> >
> > > > just to give you an idea of which segments of surface have equal area
> > > > in either case correspond, I
> >
> > > > have put in the following sketch.
> >
> > > >http://i50.tinypic.com/2bpmbk.jpg
> >
> > > > Consider the following:
> >
> > > > The area enclosed between two axially separated planes i.e., axial
> > > > difference delZ  sliced apart  in case of sphere (commonly known as
> > > > a spherical segment)
> >
> > > > AND
> >
> > > > the annular area between radially cut/separated cylindrical shells
> > > > separated by delR along radius in case of the pseudosphere somewhat
> > > > looking like a frustum of cone if meridian curvature is neglected,
> >
> > > > are equal, when delR = delZ.
> >
> > Narasimham, I did not see this equation here on first read.
> >
> > I see from the sketch that Del Z an arc on the sphere is about 1/3 the
> > arc of Del R
> > the arc on the pseudosphere.
> >
> > It was my fault, Narasimham that I was wavering over what I wanted and
> > involved
> > area, when the question I seek does not need area at all. I was
> > troubled with finding
> > a precision clear "question to pose" and was distracted into thinking
> > that area was
> > involved.
> >
> > My question should have been this --
> >
> > How much of the arclength on a pseudosphere coincides in curvature
> > with a arclength
> > segment on a great-circle where the radius of pseudosphere and sphere
> > are identical?
>
> There are two curvatures, k1 meridional and k2 circumferential. Gauss
> curvature K = k1*k2.
> If you are referring to k1 alone alone, it does not give full picture.
> Never mind the lone curvature.
>
> My sketch does not give zones where radii are equal. The sketch is
> drawn to scale.
> You can see that There are two curvatures k1 meridional and k2
> circumferential. Gauss curvature K = k1*k2.
> The radius has decreased and width has increased keeping area same
> while the segment is going to a pseudospherical one from spherical
> surface.
>
> Exact dimensions are given below:
>
> Sphere meridian end points (z, r ) for shaded area are
> start ( 0.4, 0.916515 ) ; end ( 0.6, 0.8 )
> Arc length of meridian = 0.231984
> Area = 2 pi 1 0.2 (del Z)= 1.25664
>
> Pseudosphere meridian end points (z, r ) for shaded area are
> start ( 0.650284, 0.4) ; end ( 1.31264, 0.2 )
> Arc length of meridian = 0.693147, about thrice that of sphere.
> Area = 2 pi 1 0.2 (del R) = 1.25664
>
> > So, Narasimham, how much of Del R can be fitted onto the sphere's
> > great-circle?
> > Can all of that Del R arc be equal to a great-circle in curvature?
>
> Cut up sphere meridian into 10 approximate conical sectors and again
> circumferentially
> take 3 cuts on all the segments. Now cut the, pseudosphere also
> into 30 conical sectors on meridian. These represent differential
> areas, they can be matched
> in area by placing them side by side.
>

I do not want to match areas. I am sorry I ever brought up the subject
of area.

I want to match arcs. So that a arc of the pseudosphere fits perfectly
onto a
arc of a great circle. So if I cut an arc segment from the
pseudosphere like this
")" and then on the sphere cut an arc of this ")" that the two arcs
match and
can be fitted inside one another, or one can replace the other.

So, Narasimham, from your sketch are there any arcs on that
pseudosphere
line you have drawn that can replace an equal arc on the great-circle
of the
sphere you have drawn in the sketch?

And are you saying, Narasimham, that I can retrieve several arcs from
lines
on the pseudosphere that match arcs of great-circles on the sphere? By
match,
I mean they are identical and can replace one another. By match, I
mean that
if I cut a 60 degree arc from a sphere and another 60 degree arc from
a identical
sphere of identical radius that I can replace one arc from sphereA
onto sphereB.
That the two arcs ( and ( can replace one another.

Please forget that I ever mentioned area. I simply want to know
whether any arcs
on a pseudosphere of 1 unit radius matches another arc on a great-
circle of a
sphere of 1 unit radius.

Narasimham, from your sketch, it appears that some arcs of the
pseudosphere match
arcs on the sphere. Is that true? Is there more than one arc on the
pseudosphere that
matches the arcs on the great circle of the sphere?

> Never mind about the curvature that way.
>
> In Bending, it is the lengths and angles between lines on surface
> which remain constant.
> Also curvature of lines in tangent plane known as geodesic curvature
> is invariant.
>
> These are creatures of the first fundamental form of surface theory in
> classical differential
> geometry.A study of the same by you , known as Isometry or in other
> words compound Bending will be useful.
> It is possible to apply one surface of constant K onto another,
> without tearing or crumpling up.
>
> Gauss Egregium theorem states that K = k1 * k2 is constant in such
> cases.
> But not a wholesale switch from +1 to -1 is permitted in isometry.
>
> Isometry( preserving angles,lengths ) and equal area mappings are two
> different
> and distict things, not to be confused but regarded as different.
>
> If area is changed as a rubber sheet, then the Gauss-Bonnet theorem
> gives what the
> restrictions are....
>
> > > > It is not difficult to derive these as the areas are part of surfaces
> > > > of revolution.
> >
> > > > in case of sphere/pseudosphere of radius / pseudoradius a,
> >
> > > > full rotated Areas are = 2 pi a delZ and  2 pi a delR respectively.
> >
> > > > If the polar angle difference is same and equal to delTheta,
> >
> > > > Area/a = delTheta * delZ = delTheta * delR is the condition for equal
> > > > area mapping.
> >
> > > > Hope this helps you to determine what or how much to "Roll" in each
> > > > case for each equal area differential shell
> >
> > > > surface elements.
> >
> > > > Narasimham
> >
> > I could still ask for an area with a "roll" function involved. So how
> > much of
> > Del R has the identical curvature of the sphere, and then to roll the
> > sphere
> > at that identical-arc-site on the pseudosphere, roll it around the
> > pseudosphere
> > spine and to obtain the area involved. Of course, I would have to
> > multiply by
> > 2 because of the southern hemisphere of the pseudosphere.
> >
> > Narasimham, is the maximum equal-curvature of sphere with pseudosphere
> > that
> > of 36 degrees arc? Your sketch seems to suggest it is larger than 36
> > degrees arc.
> >
> > P.S. of course, the answer could be 0 arc on the sphere is identical
> > to the curvature
> > of any and all arcs on the pseudosphere.
> >
> > Archimedes Plutoniumwww.iw.net/~a_plutonium
>
> Hope you heed my suggestion to study classical differential geometry.
> Good Luck in that.

I wish I had taken differential geometry in my youth. It is a subject
far more
useful than many algebra subjects.

>
> Narasimham


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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