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Orchid XP v8 wrote:
>>> What about xyz = the ratio of any circle's circumference to its
>>> diameter?
>>
>> Define "circle".
>
> The set of all points at distance r from the point c.
>
>> Now define "distance".
>
> I actually can't find a definition for how to do this in a non-Euclidean
> space.
Oh, I forgot. Now you have to also define the length of a curved line with
no endpoints. :)
> Well, yes... I'm sure somebody somewhere has long since worked all this
> out in excruciating detail. ;-)
Well, yes.
--
Darren New, San Diego CA, USA (PST)
Forget "focus follows mouse." When do
I get "focus follows gaze"?
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Stephen wrote:
>
> What about xyz = the ratio of any circle's circumference to its diameter?
>
The main problem here is that xyz is not a constant, but is rather a
function. In spherical and hyperbolic spaces it can be defined as a
function of the radius so you need to say something like:
xyz(r) = the ratio of the circumference to the diameter of a circle of
radius r
If your space doesn't have a constant curvature, then it's a function of
yet more parameters.
The main issue isn't only that pi has a pre-existing meaning, it's that
the value in these non-euclidean spaces isn't even a constant.
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Darren New wrote:
> Now define "distance".
I thought the underlying assumption in this thread was that we're
dealing spaces equipped with a metric tensor, do you automatically get a
definition of distance in the standard manner. How does this require
extra care to define?
I suppose you could argue that circles are a bit trickier to define, but
I think the normal one will work well, and should even be well-behaved
on genus-0 manifolds, or locally on any smooth manifold (like spherical
or hyperbolic spaces).
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Kevin Wampler wrote:
> should even be well-behaved
> on genus-0 manifolds
I take that back, it's obviously not quite well behaved on a sphere
(under some definitions of well-behaved) since there are no circles of
sufficient large radii.
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>> should even be well-behaved on genus-0 manifolds
>
> I take that back, it's obviously not quite well behaved on a sphere
> (under some definitions of well-behaved) since there are no circles of
> sufficient large radii.
Any possible line segment in elliptic geometry can be used as a circle
diammeter. (Since lines that are "too large" to define a circle won't
fit into the space in the first place.)
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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Kevin Wampler wrote:
> I thought the underlying assumption in this thread was that we're
> dealing spaces equipped with a metric tensor, do you automatically get a
> definition of distance in the standard manner. How does this require
> extra care to define?
>
> I suppose you could argue that circles are a bit trickier to define, but
> I think the normal one will work well, and should even be well-behaved
> on genus-0 manifolds, or locally on any smooth manifold (like spherical
> or hyperbolic spaces).
That's a whole crapload of jargon, right there.
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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Orchid XP v8 wrote:
>
> Any possible line segment in elliptic geometry can be used as a circle
> diammeter. (Since lines that are "too large" to define a circle won't
> fit into the space in the first place.)
>
Yes, obviously, so long as you don't define line segments so that a line
is allowed to overlap with itself. But I defined xyz as a function of
the *radius* of a circle, not as a function over line segments.
Defining circles so that they work for all radii under this definition
requires a touch more care (although it is possible).
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Orchid XP v8 wrote:
>
> That's a whole crapload of jargon, right there.
>
Those are all very basic terms for dealing with non-Euclidean spaces.
Since we're talking about non-Euclidean spaces, I don't think it's
uncalled for.
That said, you do have something of a point there, so I'll rephrase:
I thought we were assuming that the spaces we were talking about come
with a pre-defined notion of distance, so why is there a need to
explicitly define it ourselves?
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>> That's a whole crapload of jargon, right there.
>
> Those are all very basic terms for dealing with non-Euclidean spaces.
> Since we're talking about non-Euclidean spaces, I don't think it's
> uncalled for.
Not uncalled for, no. I'm just impressed that there exist people who
actually know WHAT THE HELL THIS STUFF MEANS!
> That said, you do have something of a point there, so I'll rephrase:
>
> I thought we were assuming that the spaces we were talking about come
> with a pre-defined notion of distance, so why is there a need to
> explicitly define it ourselves?
Yeah, that was basically my assumption too.
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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Kevin Wampler wrote:
> I thought we were assuming that the spaces we were talking about come
> with a pre-defined notion of distance, so why is there a need to
> explicitly define it ourselves?
I think I was more making the point that "distance" isn't a simple
definition when you're talking about space that curves in different
directions in different places.
Indeed, isn't it possible to have spaces where the distance from here to
there is different than the distance from there to here?
--
Darren New, San Diego CA, USA (PST)
Forget "focus follows mouse." When do
I get "focus follows gaze"?
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