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From: Archimedes Plutonium on 8 Feb 2010 03:33
Alright, I went to this website:

http://en.wikipedia.org/wiki/File:Tractrix.png

And then constructed a circle of equal radius to the picture shown.
Then I moved the circle around on the tractrix to see if the arc of
the
circle matches the arc in the tractrix. It does not match very close
to the cusp, but it does match about 1/2 ways from the cusp graph
square
into half of the next graph square. From the squares of 0,3.5 to 0,
2.5
looks like an alignment of arcs of the tractrix with circle. But the
problem
with this method is that the lines are too wide to really see any
precision.
But this method does lend credence to the idea that an arc of the
tractrix
matches an arc of the circle (sphere).

I have looked through the literature to see if there are any theorems
in
geometry that would immediately and automatically eliminate the idea
of
a arc of the tractrix never able to match an arc of the sphere
involved.
There is none.

However, there is a fact or quasi theorem in math that suggests there
is
an arc to match in both tractrix and sphere. The fact is Euler's
identity
of sine and cosine that yields e^(i x 2pi) = 1 wherein the pi relates
to the
sphere and "e" relates to the tractrix. So this identity suggests
there is
a matching arc in both tractrix and associated great circle of sphere
involved.

The idea behind the matching is that the sphere has a constant-arc-
curvature
but that the tractrix has what can be called a collection of sequental
varying arc curvatures wherein one of those arcs fits an associated
arc on the circle.


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 8 Feb 2010 15:01
On Feb 8, 1:33 pm, Archimedes Plutonium
<plutonium.archime...(a)gmail.com> wrote:
> Alright, I went to this website:
>
> http://en.wikipedia.org/wiki/File:Tractrix.png
>
> And then constructed a circle of equal radius to the picture shown.
> Then I moved the circle around on the tractrix to see if the arc of
> the
> circle matches the arc in the tractrix. It does not match very close
> to the cusp, but it does match about 1/2 ways from the cusp graph
> square
> into half of the next graph square. From the squares of 0,3.5 to 0,
> 2.5
> looks like an alignment of arcs of the tractrix with circle. But the
> problem
> with this method is that the lines are too wide to really see any
> precision.
> But this method does lend credence to the idea that an arc of the
> tractrix
> matches an arc of the circle (sphere).
>
> I have looked through the literature to see if there are any theorems
> in
> geometry that would immediately and automatically eliminate the idea
> of
> a arc of the tractrix never able to match an arc of the sphere
> involved.
> There is none.
>
> However, there is a fact or quasi theorem in math that suggests there
> is
> an arc to match in both tractrix and sphere. The fact is Euler's
> identity
> of sine and cosine that yields e^(i x 2pi) = 1 wherein the pi relates
> to the
> sphere and "e" relates to the tractrix. So this identity suggests
> there is
> a matching arc in both tractrix and associated great circle of sphere
> involved.
>
> The idea behind the matching is that the sphere has a constant-arc-
> curvature
> but that the tractrix has what can be called a collection of sequental
> varying arc curvatures wherein one of those arcs fits an associated
> arc on the circle.

Why is it so exciting? Every non- circular curve has a variable
curvature.
Here what you are seeking does not even agree in the sign of
curvature.You
are finding nothing new here.

As a matter of fact,the radii of curvatures of a tractrix are too well
known:

R1 = - a Cot(ph), R2 = a Tan(ph) where ph is tangent slope. a is
pseudoradius.
R1 < 0 anticlastic. So at ph = pi/4, curvature is same, upto sign.

Instead of the narrow grid you asked that serves no purpose, I have
given a sketch
of catenoid which is the evolute of Tractrix.You can unwind a string
on a rigid
catenary to get pseudosphere meridian = Tractrix. You get an exact
curvature match further
down when ph = pi/4 or 45 degrees. But so what?

http://i45.tinypic.com/xeitcn.jpg

Narasimham




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