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From: Archimedes Plutonium on 6 Feb 2010 16:33


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?

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?

> >
> > 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 Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
From: gudi on 6 Feb 2010 19:03
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.

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.

Narasimham


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