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hyperbolic lines

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  • Francois Rideau
    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]
    • Jim Parish
      ... 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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        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
      • Francois Rideau
        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:
          >
          > 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]
        • Ken Pledger
          ... 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:
            >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".)....


            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
            "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.
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