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From: Archimedes Plutonium on 26 Jan 2010 15:15


Archimedes Plutonium wrote:
> Enrico wrote:
> (snipped)
> >
> > Don't know of any formulas or equations, but it brings
> > to mind stuff like projective geometry or descriptive
> > geometry - its been decades since I studied those.
> >
> > Try starting here:
> >
> > http://mathworld.wolfram.com/ProjectiveGeometry.html
> >
> > I had to get a ball to try to see what you're doing.
> > No luck. Are the sides of the rectangle great circle
> > segments or are they following latitude and longitude
> > lines? Are these the wrong questions?
> >
> >
> > Enrico
>
> Hi, thanks for tackling the question. Sorry that I am not
> as clear as to the demonstration.
>
> I should explain it in a different way since it is confusing as to how
> I am reversing the concavity of the sides of a triangle on a sphere.
>
> So I should render the question-problem as this: that we have a
> triangle embedded inside a rectangle of a globe whose latitude and
> longitude
> is 36 by 36 degrees. So I want to know if I reverse the concavity of
> that
> embedded triangle (the two triangles formed by the 4 vertices of the
> rectangle).
> Whether that triangle is the upper limit of constructing a reverse
> concavity triangle.
>
> Enrico, I suppose another way of stating the problem is that given a
> globe
> and given a rectangle of 36 by 36 degrees. Now handed a "saddle shaped
> object"
> that somehow (some math term meaning the reverse analogy of that
> sphere-globe)
> Here the question is, does the globe rest into the saddle at only a 36
> by 36 patch?
>
> I am trying to figure out, since the concavity of a Elliptic geom
> triangle is concave
> outwards and a Hyperbolic geom triangle is concave inwards. I am
> trying to figure
> out the Upper bound of constructing triangles between the two
> geometries. Somewhere
> in the construction, the concave inwards can no longer handle the same
> 3 point vertices
> that the elliptic triangle handles because the sides do not meet at
> the vertices.
>
> I am guessing that it is a 36 by 36 degree latitude and longitude
> rectangle as the upper bound.
>
> I am hoping that David Bernier has the answer on the tip of his tongue
> with the proper
> trig equation.
>

Sorry to everyone on this problem or demonstration, especially Enrico.

I should have started the problem by using only Euclidean geometry and
then
scaled up to Elliptic and Hyperbolic. That way I could explain it with
ease.

In Euclidean geometry we draw a triangle, say a right triangle. Now we
outfit that
triangle with curved sides rather than with straight line segments.
Depending on the
curvature of those sides we look for reversing the concavity of those
curved sides.
The question here in Euclidean geometry is the upper bound or upper
limit to having
curved sides concave outward of a triangle and then reversing the
concavity, yet still
maintain the three vertices.

So it depends on the amount of curvature of the sides, and the size of
the triangle,
but what I wish to seek is some equation that says the limit at which
it is impossible
for the reverse concavity to still maintain it being a triangle.

So I should have started in Euclidean geometry and then generalized
over into Elliptic
and Hyperbolic.

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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