Re: Poll on *Really* Wide Angle Lenses


"David Littlewood" <david@xxxxxxxxxxxxxxxxxx> wrote in message
news:ONvZ9BC+mJCDFw23@xxxxxxxxxxxxxxxxxxxx
> In article <su6dnQGjY9KSDpXeRVn-tQ@xxxxxxxxxxx>, Nostrobino
> <not@xxxxxxxxxx> writes
>>
>>>
>>> If you think navigation is a more appropriate tool than mathematics for
>>> discussing geometric optics and photometry, then we will just have to
>>> agree to differ.
>>
>>No, not at all, but I was replying to your "Only straight lines can be
>>parallel" assertion. I have no idea whether there is some requirement in
>>mathematics that this must be so. If your wife, a mathematician, says it
>>is
>>so then I'm perfectly willing to accept that it is so *in mathematics*,
>>but
>>then that must be a special or at least limited case. The first definition
>>for "parallel" in my dictionary is simply "Being an equal distance apart
>>at
>>every point." And again, lines of latitude are commonly called parallels
>>and
>>have been for centuries. Navigators do not appear to have any problem with
>>that usage.

BTW, David, my original choice of words was probably not the best. When I
wrote:
"You are trying to apply some special definition which just does not fit the
general case. Lines of latitude have been called parallels for centuries,
still are today and I'll bet they always will be. If your wife were a
navigator instead of a mathematician I'm sure she would understand this
better."
--I did not mean to suggest that your wife's understanding was deficient,
certainly not within her own sphere of expertise. I should have said "would
understand this differently" rather than ". . . better." I apologize if any
misunderstanding resulted from my carelessness in language.


>>
> Think about the definition of Rieman 2-space. "Parallel lines never meet"
> is replaced by "Parallel lines always eventually meet". If lines of
> latitude were "parallel", this would become meaningless.

I have to tell you I just don't have the foggiest notion of what "Rieman
2-space" means. It's been years since I studied any flavor of geometry, and
that term I don't recall ever having seen prior to this thread.

I never even heard or saw the term "2-space" before a few days ago. I gather
it refers to the treating of a sphere's three-dimensional surface as though
it were two-dimensional, for purposes of geometric calculation.

It does all sound very interesting, but at present leaves me scratching my
head.


>
> Also, bear in mind that (as I said to someone yesterday) we were talking
> about the projection of straight lines on a a spherical surface in
> 2-space. If they are not great circles, they will not be recorded as
> straight lines (absent some processing algorithm).

Can you refresh my memory: why were we doing that? I don't really have any
opinion, or anything to say, about "the projection of straight lines on a a
spherical surface in 2-space." Is this in connection with the image formed
on the retina? Because my opinion about that has been, and remains, that the
exact form or nature of the retinal image is of little if any importance
beyond the fact that it is the source of signals along the optic nerve that
the brain sorts out to create visual perception. All the retinal image has
to be is some form that the brain can make use of in that way, in other
words. I presume that the image of a square, say, must be at least somewhat
square-like on the retina, but I doubt that it has to follow any rules of
spherical geometry.


>>
>>>>
>>>>Why should separation or perpendicularity be "not definable" on curved
>>>>parallel lines? The arc still has a tangent, doesn't it? Separation
>>>>would
>>>>be
>>>>simply the distance between inside and outside tangents, where *they*
>>>>are
>>>>parallel (i.e., at the closest point on one arc to any point on the
>>>>other).
>>>>I don't see a problem there, and I'm not a mathematician.
>>>>
>>> OK, it may be definable, but they are still not parallel.
>>
>>The dictionary disagrees with you, David. In the American Heritage
>>Dicitonary, Second College Edition, "parallel" definition 2.d. is
>>"Designating curves or surfaces everywhere equidistant."
>>
> If this were correct, then 2.5 millennia of Euclidean geometry has been
> sadly misinformed. This kind of confusion occurs when one has a scientific
> argument using a term rigorously defined in science and used any which way
> in non-science - if one side tries to use the non-science meaning, there
> can be no useful discussion.

The same dictionary also gives a definition I know you will like better:
2.a. "Designating two or more straight coplanar lines that do not
intersect." On the other hand, of course, that is Euclidean and violates the
"Rieman 2-space" rule you appear to be endorsing above. Ergo, when you speak
of "a term rigorously defined in science" in this connection it's a term
that you yourself are using in at least two mutually exclusive ways.

N.


.

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