Re: Poll on *Really* Wide Angle Lenses

In article <EIWdncJjapqqQ5XeRVn-2Q@xxxxxxxxxxx>, Nostrobino <not@xxxxxxxxxx> writes 

"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.

No problem - I did not take it as a personal attack. However, I still assert that mathematics is the right tool for discussing geometry. 


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.

"2-space" simply means 2 dimensions. Rigorously. You are not allowed to even imagine that there is a third. However, there is no reason to think that, for a hyper-dimensional being, he would see our restricted world as a plane. It could be a sphere, or some other surface. The important thing is that we, the earthlings, don't even imagine the "other" dimensions. It should be quite easy for us earthlings, as until flight (and ignoring miners and mountaineers) we were mostly constrained thus. 

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.

Well, you got it about right. A straight line is the shortest distance between two points, which, on a spherical 2-space, even navigators (ahem, sorry) acknowledge means a great circle. Any other route (including a line of latitude other than the equator) is a curve. 


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.

I thought (and I accept it got so confusing I may be wrong) we were talking about rectilinear projection. If a sensor is a spherical surface, and ignoring brain or other processor algorithms, then only a great circle line on that sphere will be recognised as "straight"; anything else will record as a curve. I've almost forgotten why this mattered though... 



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.

Classical (Euclidean) geometry is based on a series of postulates by Euclid (Greek philosopher ca. 300BC). They are not something which can be proved, they just have to be assumed. The final one was "parallel lines continue to infinity and never meet, but continue to be the same distance apart". Rieman and Lobachevsky (19thC German and Russian respectively IIRC) each postulated an alternative: (a) parallel lines always meet eventually, and (b) parallel lines continuously diverge. Perfectly credible geometries result, and the former happens to coincide with what we experience here on earth: a straight line is a great circle, step a few feet away and generate another great circle in the same direction, produce them and they will meet half way round the earth. Straight lines, parallel, inevitable meet.

However, this is not found in high school geometry books, so lexicographers can be forgiven for not knowing it. I'm sure you have experience yourself of general dictionaries being completely in the dark for higher technical matters.

BTW, I guess a more general definition of parallel would be "two or more straight lines for which a line perpendicular to one of the lines at any point is also a perpendicular to the other(s). I must ask my wife if she agrees, but we just shared a bottle of wine with dinner and I'm not sure how receptive she would be!

David
--
David Littlewood
.

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