- 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] - Francois Rideau asked:
> In hyperbolic geometry, how are named two lines which share one end on the

In my experience, they are called "parallel". (Pairs of lines which do not

> horizon?

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.

Friendly

Francois.

On 10/31/07, Jim Parish <jparish@...> wrote:

>

> Francois Rideau asked:

> > 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] - At 9.59 AM -0500 31-10-07, Jim Parish wrote:
>Francois Rideau asked:

Those names are used quite widely, but

>> 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".)....

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

"horoparallel" plus several other helpful

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.