## hyperbolic lines

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• Dear friends In hyperbolic geometry, how are named two lines which share one end on the horizon? Friendely Francois [Non-text portions of this message have
Message 1 of 4 , Oct 31, 2007
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Dear friends
In hyperbolic geometry, how are named two lines which share one end on the
horizon?
Friendely
Francois

[Non-text portions of this message have been removed]
• ... In my experience, they are called parallel . (Pairs of lines which do not intersect, even on the horizon, are ultraparallel .) Jim Parish ... SIUE Web
Message 2 of 4 , Oct 31, 2007
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> In hyperbolic geometry, how are named two lines which share one end on the
> horizon?

In my experience, they are called "parallel". (Pairs of lines which do not
intersect, even on the horizon, are "ultraparallel".)

Jim Parish
-------------------------------------------------
SIUE Web Mail
• Thank you, Jim. I thank it was just the opposite, that is to say, parallel when they don t intersect and ultraparallel when they share one end on the horizon.
Message 3 of 4 , Oct 31, 2007
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Thank you, Jim.
I thank it was just the opposite, that is to say, parallel when they don't
intersect and ultraparallel when they share one end on the horizon.
Friendly
Francois.

On 10/31/07, Jim Parish <jparish@...> wrote:
>
> > In hyperbolic geometry, how are named two lines which share one end on
> the
> > horizon?
>
> In my experience, they are called "parallel". (Pairs of lines which do not
> intersect, even on the horizon, are "ultraparallel".)
>
> Jim Parish
> -------------------------------------------------
> SIUE Web Mail
>
>
>

[Non-text portions of this message have been removed]
• ... Those names are used quite widely, but there seems to be no universal standard. Recently I gave a lot of thought to changing the terms I use in teaching,
Message 4 of 4 , Oct 31, 2007
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At 9.59 AM -0500 31-10-07, Jim Parish wrote:
>> In hyperbolic geometry, how are named two lines which share one end on the
>> horizon?
>
>In my experience, they are called "parallel". (Pairs of lines which do not
>intersect, even on the horizon, are "ultraparallel".)....

Those names are used quite widely, but
there seems to be no universal standard.
Recently I gave a lot of thought to changing the
terms I use in teaching, and looked up some
historical background. You may like to see part
of the resulting list for (a) lines which meet on
the ideal conic, and (b) lines which meet outside
the ideal conic.

c. 1817-1820?, Bolyai (& Szasz?): (a) asymptotic parallel.
(Cf. Bonola, "Non-Euclidean Geometry," p. 97.)

No later than 1831, Gauss: (a) parallel. (Cf. Bonola pp. 67-68.)

Between 1899 and 1943 (the edition I have),
Hilbert: (a) parallel, (b) neither intersect
nor are parallel.
("Foundations of Geometry" Appendix III, ยง IV)

1914, Sommerville: (a) parallel, (b) non-intersecting.
("Non-Euclidean Geometry" pp. 29-30 and 40-41)

Between 1942 and 1965, Coxeter: (a) parallel,
(b) ultra-parallel (with the hyphen).
("Non-Euclidean Geometry" pp. 175, 188.
It appears that he may be introducing
"ultra-parallel" as a new term.)
1961, Coxeter: (a) parallel, (b) ultraparallel
(without the hyphen) or hyperparallel.
("Introduction to Geometry" 1st edition, pp. 188-9, 268)

1950, Forder: (a) limit ray.
("Geometry" p. 82)

Between 1963 and 1972, Eves: (a) parallel, (b) hyperparallel.
("Survey of Geometry" p. 292)

From the 1960s onward, other text-books used
various names; but the one I think most notable
is:
1975, George E. Martin: (a) horoparallel, (b) hyperparallel.
("The Foundations of Geometry and the
Non-Euclidean Plane" pp. 339, 341)

I have e-mailed Prof. Martin and checked
that he did in fact invent the term
neologisms in his book. The orthogonal
trajectory of a set of HYPERparallel lines is a
HYPERcycle (another name for the equidistant
curve). Likewise the orthogonal trajectory of a
set of HOROparallel lines is a HOROcycle (a
long-established term - the version "horicycle"
goes back to Lobachevsky).

In teaching I find it quite helpful to be
able to retain Euclid's definition of parallels
as coplanar lines which don't meet, and simply
subdivide such parallels into horoparallels and
hyperparallels. (Incidentally, I prefer the pure
Greek "hyperparallel" to the Latin/Greek hybrid
"ultraparallel".)

So the simple answer to Francois's
question is that there's no simple answer. :-)

Ken Pledger.
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