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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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Darren New wrote:
> 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.
Sure. I get that.
> Indeed, isn't it possible to have spaces where the distance from here to
> there is different than the distance from there to here?
No. The definition of "metric" demands that it is symmetric (and
reflexive, and transitive).
You can, of course, define a function that represents some sort of
"nearness" which never the less lacks this property...
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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Darren New wrote:
>
> 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.
I suppose I normally view distance in Euclidean space from the same
definition that applies to non-Euclidean spaces, rather than the
sqrt(dx^2 + dy^2) form, so I didn't really consider this. You're right
though, if you're starting from the pythorgean theorem view of distance
it does bear some thinking about how it generalizes to the non-Euclidean
space.
That said, I'm not sure it's necessary to actually understand the proper
definition of distance in order to talk about circles in other spaces --
particularly if we limit ourselves to spherical and hyperbolic spaces
which are more or less easy to visualize.
> Indeed, isn't it possible to have spaces where the distance from here to
> there is different than the distance from there to here?
Sure, but the standard interpretation of "non-Euclidean space" that I'm
aware, ie "Riemannian manifold", uses a definition of distance which
doesn't allow such things. All distances are proper distances in that
they obey symmetry, the triangle inequality, etc. I suppose, however,
that you could argue that "non-Eucliean" should be colloquially taken to
include pseudo-Riemannian spaces too (so as to incorporate Minkowski
spaces), but even here you have a pretty nice definition of distance,
although "circles" become a bit nastier to talk about.
It's possible other people mean different things when they say
non-Euclidean space though, so I don't want to pretend that it's the
only way to read that term. Still, that's what had interpreted to be
implicit in the conversation.
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Orchid XP v8 wrote:
>
> and transitive
>
I don't think that's what you intended to say. You probably meant to
say "triangle inequality".
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>> and transitive
>
> I don't think that's what you intended to say. You probably meant to
> say "triangle inequality".
Actually I meant that d(x,y)=0 implies x=y, but even so, that's not the
right term...
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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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.
>
In how many dimensions? Imaginary or real?
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