From: hporopudas@tnclus.tele.nokia.fi
Subject: Elliptic, hyperbolic and plane geometries
Organization: Nokia Telecommunications.
Date: Thu, 1 Oct 1992 11:01:21 GMT


  About elliptic geometry and hyperbolic geometry with aid of algebra


      I refer here to my article 'Surface Algebras'
 <1992Aug18.160700.1@tnclus.tele.nokia.fi>. This was first published in the end
 of the year 1991 in sci.math.
   I have here two details taken from it: parallelogram on sphere (elliptic
 geometry) and parallelogram on pseudo-sphere (hyperbolic geometry).
   Definition of the object called parallelogram is given in above mentioned
 article (and also in abstract of this article, which is in the end of this
 writing).
   Normal plane formulas for plane parallelogram is achieved when for example
 radius of sphere R or radius of pseudosphere S increases without limits.


               Oulu  -  Finland  1.10.1992

               Hannu, K. J. Poropudas
               Vesaisentie 9E, 90900 Kiiminki, Finland.
               (hporopudas@tnclus.tele.nokia.fi)


 THE ALGEBRA OF GENERAL ANGLE COMPLEX NUMBERS


 SUM OF TWO GENERAL ANGLE COMPLEX NUMBERS:

  _    _                                  _
  T  + T  is general angle complex number T  , length of which T  < 2*pii*R is:
   1    2                                  x                    x

      2                     2                             2
   tan ( T  / ( 2 R )) = tan  ( ( T  + T  ) / ( 2 R )) cos ( ( P - P )/ 2 ) +
          x                        2    1                       2   1

                            2                             2
                       + tan  ( ( T  - T  ) / ( 2 R )) sin ( ( P - P )/ 2 )
                                   2    1                       2   1

 rotation angle  P  + A   ( assume P  => P  ) where
                  1                 2     1

tan( A-(P - P )/2 )=tan((P - P )/2) tan((T - T )/( 2R ) )/ tan((T +T )/( 2 R ) )
         2   1            2   1           2   1                  2  1

 P   and  P  are rotation angles measured from the fixed reference direction
  1        2
 which starts from origin.

 ( In case  P  > P   the rotation angle is  P   + A  )
             1    2                          2
 Now we have to suppose

 T  < pii*R ja  T  < pii*R , ( pii = 3.1415927... radians).
  1              2
 because of the used reference (Ayres 1954, page 148) for calculations.
   Diagonals which have length more than 2*pii*R are treated so that whole
 multiples of 2*pii*R is subtracted away from the preliminary arc length under
 study. By this way achieved general angle complex number is said to be
 equivalent with the primary one.
   In the case where lengths of two directed geodesic lines are < pii*R
 and > pii*R, where pii = 3.14159... radians and R is the radius of the
 sphere surface, one pair of opposite sides of the parallelogram intersects
 each others. The sum in this case can be formed alternatively so that one
 transfers smaller directed geodesic line starting from the South-Pole. This
 tranfer is done along the extension part of this same directed geodesic line.
 The part of other directed geodesic line, which exteds over the South-Pole,
 and this transferred part is used as sides of the new parallelogram. Now the
 sum diagonal is formed same way as described in earlier parts with aid of
 geodesic circles. The length of the preliminary sum diagonal is length of
 this new diagonal + pii*R .
   When both sides of the parallelogram on the sphere surface are > pii*R,
 then sides and transferred sides of the parallelogram intersects. In this case
 sum is formed alternatively from the extension parts of the sides over South-
 Pole. Sum diagonal's length of this new parallelogram is further extented by
 an amount of pii*R.
   In the case when one side has length equal to pii*R the parallelogram is
 constructed with aid of properties of angles between diagonals and sides.
 Now parallelogram is symmetric in respect to both diagonals due to symmetry
 of sphere surface.
   On points where tan-function is singular, general definition of
 parallelogram (and properties which are led from it) on the surface is used.
 All other special cases like special points of trigonometric functions are
 treated similarly.
   This algebra seems to be useful, because lengths of general angle numbers
 are allowed to variate between 0 and +infinite.

   (The general method of constructing the sum diagonal holds also in these
 special cases of "ugly parallelograms", but one may think that these cases
 possibly contradicts with the physical interpretation which I gave. These
 intersecting and other cases must, of course, be confirmed also by the aid of
 some physical phenomena. I think that applications depends on how successful
 this interpretation is.)


 SUBTRACTION OF TWO GENERAL ANGLE COMPLEX NUMBERS:

 CASE A:
  _    _                                   _
  T  - T   is general angle complex number T  ,length of which T  < 2*pii*R is:
   2    1                                   y                   y

    2
 tan ( T  / ( 2 R ) )  =
        y

       2                              2                         2
= ( sin  (( T + T ) / ( 2 R ) ) + sin  (( T  - T )/ ( 2 R )) cot ((P -P )/2) )/
             2   1                         2    1                   2  1

        2                              2                         2
 / ( cos  (( T + T ) / ( 2 R ) ) + cos  (( T  - T )/ ( 2 R )) cot ((P -P )/2) )
              2   1                         2    1                   2  1
               _       _
 angle between T  and  T  is w , (angle between tangent lines).
                y       1

 tan w =

( cos((T -T )/(2R))/cos((T +T )/(2R))-sin((T -T )/(2R))/sin((T +T )/(2R)) ) *
        2  1              2  1              2  1              2  1

* cot (( P  - P )/ 2 ) /
          2    1
                            2
/ ( 1 + sin((T  -T )/ R) cot ( ( P -P )/ 2 )/ sin((T +T )/ R )  )
              2   1               2  1              2  1
               _                  _
 angle between T  and transferred T   is q,
                y                  2
 (transfer with aid of the parallelogram on the sphere).

 Rotation angle is  P  + q  (assume  P => P  ) .
                     2                2    1
 ( if P   > P    , then rotation angle is  P  - q  )
       1     2                              2

 tan q =

( cos((T -T )/(2R))/cos((T +T )/(2R))+sin((T -T )/(2R))/sin((T +T )/(2R)) ) *
        2  1              2  1              2  1              2  1

* cot (( P  - P )/ 2 ) /
          2    1
                            2
/ ( 1 - sin((T  -T )/ R) cot ( ( P -P )/ 2 )/ sin((T +T )/ R )  )
              2   1               2  1              2  1

 where * is ordinary multiplication and  T  < pii*R ja  T  < pii*R .
                                          1              2
 (Ayres 1954, page 148).

 CASE B:
  _    _                                   _
  T  - T   is general angle complex number T  ,length of which T  < 2*pii*R is:
   1    2                                   g                   g

  is equal T  defined above.
            y
               _       _
 angle between T  and  T  is same q as above (angle between tangent lines).
                g       2
               _                   _
 angle between T   and transferred T  is w.
                g                   1

 Rotation angle is  P  - w  (assume  P => P  ) .
                     2                2    1
 ( if  P   > P    , the rotation angle is  P  + w  ), where
        1     2                             2

 tan w =

( cos((T -T )/(2R))/cos((T +T )/(2R))-sin((T -T )/(2R))/sin((T +T )/(2R)) ) *
        2  1              2  1              2  1              2  1

* cot (( P  - P )/ 2 ) /
          2    1
                            2
/ ( 1 + sin((T  -T )/ R) cot ( ( P -P )/ 2 )/ sin((T +T )/ R )  )
              2   1               2  1              2  1

 where T  < pii*R and T  < pii*R, (Ayres 1954, page 148).
        1              2

         _  _                                        _       _
 PRODUCT T  T  OF TWO GENERAL ANGLE COMPLEX NUMBERS  T  AND  T  :
          1  2                                        1       2
 _  _
 T  T   general angle complex number, length of which T  T  < 2*pii*R,  and
  1  2                                                 1  2
 and rotation angle of which is  P  + P   .
                                  1    2
 (all whole multiples of 2*pii*R is subtracted away from the primary product).

          _     _                                       _      _
 QUOTIENT T  /  T  OF TWO GENERAL ANGLE COMPLEX NUMBERS T  AND T :
           1     2                                       1      2

 CASE A:
 _   _
 T / T  is general angle complex number, length of which is T  / T  < 2*pii*R,
  1   2                                                      1    2
 if T  <> 0, and rotation angle of which is P  - P  .
     2                                       1    2
 (all whole multiples of 2*pii*R is subtracted away from the primary quotient).

 CASE B:
 _   _
 T / T  is general angle complex number, length of which is T  / T  < 2*pii*R,
  2   1                                                      2    1
 if T  <> 0, and rotation angle of which is P  - P  .
     1                                       2    1
 (all whole multiples of 2*pii*R is subtracted away from the primary quotient).

 Rem. The algebra of angle complex numbers is special case (R=1) in previous
 formulas. In this case we can equivalently speak of angles and arc lengths.


 GUIDING FORMULAS:

                                                        _   _      _
   Let first geodesic triangle on the sphere have sides T , T  and T  ,
                                                         1   2      y
 length of which sides < pii*R,  (This limitation is not present on so called
                                        _      _
 "M|bius triangles"). And angle between T  and T  is P  - P  ,
                                         1      2     2    1
                                   _      _
 (assume P => P  ) , angle between T  and T  is  q .
          2    1                    2      y

 Then we have T  from the formula:
               y

  tan ( T  / ( 2 R )) = tan ( ( T  - T  )/ ( 2 R )) sin( ( w + q )/ 2 ) *
         y                       2    1

                        * csc( ( w - q )/ 2 )

 and we have w and q from the formula:


  tan( ( w + q )/ 2 ) = cos( ( T  - T  )/ ( 2 R )) sec( ( T  + T  )/ ( 2 R ) )*
                                2    1                     2    1

                        * cot ( ( P  - P  )/ 2 )
                                   2    1

  tan( ( w - q )/ 2 ) = sin( ( T  - T  )/ ( 2 R )) csc( ( T  + T  )/ ( 2 R ) )*
                                2    1                     2    1

                        * cot ( ( P  - P  )/ 2 )
                                   2    1
                                                         _   _      _
   Let second geodesic triangle on the sphere have sides T , T  and T  ,
                                                          1   x      2
 length of which sides < pii*R, (This limitation is not present on so called
                                        _      _
 "M|bius triangles"). And angle between T  and T  is  A ,
                                         1      2
               _      _                               _      _
 angle between T  and T  is q + w , and angle between T  and T  is  r ,
                1      2                               2      x
 Then we have A and r from the formulas:


  tan( ( A + r )/ 2 ) = cos( ( T  - T  )/ ( 2 R )) sec( ( T  + T  )/ ( 2 R ))*
                                2    1                     2    1

                        *cot ( ( q + w )/ 2 )

  tan(( A -( q + w ))/ 2 ) = sin(( T  - T  )/ ( 2R )) csc(( T  + T  )/ ( 2R ))*
                                    2    1                   2    1

                             *cot ( ( q + w )/ 2 )

 and we have T   from the formula:
              x

  tan( ( T  / ( 2 R )) = tan( ( T  - T  )/ ( 2 R )) sin( ( A + r )/ 2 )*
          x                      2    1

                         * csc( ( A - r ) / 2 )

 (* = ordinary multiplication).
   When we process these formulas we have the formulas in primary definitions.
 (Ayres 1954, pages 168-179, case 3). Limitations to lengths comes from
 assuptions of the same reference (Ayres 1954, page 148), where so called
 "Euler's triangles" are treated.


 PROPERTIES OF ALGEBRA OF ANGLE COMPLEX NUMBERS (R=1):

 (Suggested properties from experimental calculations, in case
  where these angle complex numbers starts from the same origin)

 (P and T measured in degrees in counter examples, othervise radians are used).

    _      _    _                         _    _      _
1.  T  + ( T  + T  )  is generally < >  ( T  + T  ) + T
     1      2    3                         1    2      3
                             _
        (counter example ):  T  = ( P , T ) = ( 43 , 125 )
                              1      1   1
         _                     _
         T  = ( 10 , 25 )  ja  T  = ( 30 , 45 )
          2                     3
         _    _
         T  + T  = ( 22.54242538 , 69.23069923 )
          2    3
         _    _
         T  + T  = ( 31.90351235 , 148.9052208 )
          1    2
         _      _    _
         T  + ( T  + T  ) = ( 32.08835756 , 165.5429008 )
          1      2    3
           _    _      _
         ( T  + T  ) + T  = ( 30.8034433 , 166.0929254 )
            1    2      3

    _    _    _                          _  _     _  _
2.  T  ( T  + T  )   is in general < >   T  T  +  T  T
     1    2    3                          1  2     1  3

         (counter example):
          _                   _                     _
          T  = ( 43 , 125 ) , T  = ( 10 , 25 )  ja  T  = ( 30 , 45 )
           1                   2                     3
          _    _
          T  + T   = ( 22.54242538 , 69.23069923 )
           2    3
          _  _
          T  T   = ( 53 , 54.5415141 )
           1  2
          _  _
          T  T   = ( 73 , 98.17477041 )
           1  3
          _    _    _
          T  ( T  + T  ) = ( 65.5424254 , 151.03796 )
           1    2    3
          _  _     _  _
          T  T  +  T  T  = ( 63.98143419 , 152.3153715 )
           1  2     1  3

    _    _     _    _
3.  T  + T  =  T  + T
     1    2     2    1
    _  _    _  _
4.  T  T  = T  T
     1  2    2  1
    _   _  _          _  _    _
5.  T ( T  T  )  =  ( T  T  ) T
     1   2  3          1  2    3
                                         _
6. There exists equivalent zero elements 0 such that
     _   _    _   _   _
     T + 0  = 0 + T = T
      1            1   1

    (for example T = n*2*pii radians, where n= 0,1,2,... and P is real number).
                                         _
7.  There exist equivalent unit elements 1 for multiplication such that
      _ _    _  _   _
      1 T  = T  1 = T
         1    1      1

    (for example T = 1 and P = n*2*pii radians, where n = 0,1,2,... )

                                  _ -1
8.  There exists inverse element  T    for multiplication such that
                                   1

         _ -1 _    _  _ -1   _
         T    T  = T  T    = 1
          1    1    1  1

    (for example  T=  1 / T   and   P = - P  )
                           1               1
                                   _
9.  There exists inverse element - T  for addition such that
                                    1
         _        _         _      _    _
         T  + ( - T  ) = ( -T  ) + T  = 0
          1        1         1      1

    (for example   T = T  and P = P  + pii )
                    1          1



      _   _        _    _                      _
10. ( T - T  ) + ( T  - T  ) is in general < > 0
       1   2        2    1

    (counter example)
     _                   _
     T  = ( 43 , 125 ) , T  = ( 10 , 25 )
      1                   2
      _   _
    ( T - T  ) = ( 209.320739 , 103.2675534 ) , q = 27.28240937,
       1   2                                    w = 166.3207391
      _   _
    ( T - T  ) = ( 342.7175906 , 103.2675534 )
       2   1
      _   _        _    _
    ( T - T  ) + ( T  - T  )  =  ( 276.0191648 , 118.4032846 )
       1   2        2    1
        _        _          _    _
11. ( - T ) + ( -T  ) = - ( T  + T )
         1        2          1    2
        _     _                     _    _                       _    _
12. ( - T ) + T   is in general < > T  - T  is in general < > -( T  - T  )
         1     2                     2    1                       1    2

    (couter example)

          - ( P , T  ) = ( 25 + 180 , 77 )  ja  ( P  , T  ) = ( 80 , 109 )
               1   1                               2    2

    First case          = ( 144.1535508 , 167.0547956 )

    Second case         = ( 143.6899819 , 62.92281022 ), q = 63.68998192 ,
                                                         w = 119.5569381
    Third case          = ( 85.44306193 , 62.92281022 )

       _   _        _  _
13. ( -T ) T  = - ( T  T  )
        1   2        1  2
       _     _     _  _
14. ( -T )( -T ) = T  T
        1     2     1  2
         _    _      _     _         _    _
15.  If  T  = T  and T   = T  , then T  = T
          1    2      2    3          1    3

      _ n _ m   _ n+m
16.   T   T   = T

      _ n    _ m    _ n-m                     _    _ m-n
17.   T   /  T    = T      , if  n  > m  and  1 /  T     , if  m > n.

        _ n  m   _ nm
18.   ( T  )   = T

       _  _   n   _ n _ n
19.  ( T  T  )  = T   T
        1  2       1   2

                  _ 0   _
20.  Definition   T   = 1  , if T < > 0

     _ -n    _     _ n
21.  T    =  1  /  T

     _ n/m             _ n                 _    n
22.  T     = m's root( T   ) = ( m's root( T ) )
     _     _    _                       _ _   _     _
23.  T = - B /  A  satisfies equation   A T + B  =  0
                          _    _    _   _
24. Angle sum in triangle T  , T  - T , T   is equal to
                           1    2    1   2
                   -1
    P  - P  + 2 tan  ( cos ((T -T )/(2R)) cot((P -P )/2) / cos((T +T )/(2R)) )
     2    1                   2  1              2  1             2  1

25. Area of triangle in case 24. is equal to

    1. On the surface of sphere
 2                   -1
R ( P  -P -pii+ 2 tan  ( cos((T  -T )/2/R) cot((P -P )/2) / cos((T +T )/2/R) ))
     2   1                     2   1             2  1             2  1

    2. On the surface of plane
                                    2
    T  T  cot ( (P -P )/2) /( 1+ cot  ((P  -P )/2) )
     1  2         2  1                   2   1
    _    _       _    _
26. T  - T  and  T  - T   has difference in rotation angles, this is equal to
     2    1       1    2
                              _    _    _    _
    the angle sum of triangle T  , T  - T  , T    .
                               1    2    1    2
                                         _   _    _
27. If a and b are scalars ( P = 0 ) and T , T  , T  are angle complex numbers
                                              1    2
    then
                  _     _     _
    1.  ( a + b ) T = a T + b T
              _             _
    2.  a ( b T ) = ( a b ) T
            _    _                         _      _
    3.  a ( T  + T  ) is in general < >  a T  + a T
             1    2                         1      2

    (counter example)
                           _                     _
          a = ( 0 , 70 ) , T  = ( 15 , 23 )  ja  T  = ( 43, 75 )
                            1                     2
      _    _
      T  + T  = ( 35.0343238 , 96.60209712 )
       1    2
          _     _
      a ( T  +  T  ) = ( 35.0343238 , 118.0217261 )
           1     2
        _
      a T  = ( 15 , 28.09980096 )
         1
        _
      a T  = ( 43 , 91.62978574 )
         2
        _       _
      a T  +  a T  = ( 34.12100781 , 118.419912 )
         1       2
               _    _                        _      _
       4.  a ( T  - T  ) is in general< >  a T  - a T
                2    1                        2      1

      (counter example)
                           _                     _
          a = ( 0 , 75 ) , T  = ( 15 , 60 )  ja  T  = ( 60 , 15 )
                            2                     1
      _    _
      T  - T  = ( 208.8015018 , 50.09948239 ) , q = 166.1984982 ,
       2    1                                   w = 52.9617606
          _    _
      a ( T  - T  )  = ( 208.8015018 , 65.58006907 )
           2    1
        _
      a T  = ( 15 , 78.53981634 )
         2
        _
      a T  = ( 60 , 19.63495409 )
         1
        _      _
      a T  - a T   = ( 210.1780054 , 65.16521967 ) , q = 164.8219946 ,
         2      1                                    w = 49.78503456
     _    _          _    _                    _  _         _  _        _  _
28.  T  / T      +   T  / T   is in gen. < > ( T  T     +   T  T  ) / ( T  T  )
      1    2   ( - )  3    4                    1  4  ( - )  2  3        2  4

      (counter example)
           _                  _                  _
           T  = ( 15 , 32 ) , T  = ( 25 , 60 ) , T  = ( 32 , 40 )  ja
            1                  2                  3
     _
     T  = ( 52 , 20 )
      4
     _    _
     T  / T   = ( 350 , 30.55774907 )
      1    2
     _    _
     T  / T   = ( 340 , 114.591559 )
      3    4
     _  _
     T  T  = ( 67 , 11.17010721 )
      1  4
     _  _
     T  T  = ( 57 , 41.88790205 )
      2  3
     _  _
     T  T  = ( 77 , 20.94395102 )
      2  4
     _  _         _  _
     T  T     +   T  T   = ( 59.2440543 , 52.93651159 )
      1  4  ( - )  2  3
                           ( 80.03347789 , 30.93749622 ) , q = 176.2482532 ,
                                                           w = 13.03347791
     _    _          _    _
     T  / T      +   T  / T   = ( -16.41718468 , 145.0343318 )
      1    2   ( - )  3    4
                                ( 359.1282378 , 84.4382591 ) , q = 174.9110933,
                                                               w = 9.128237748
       _  _         _  _        _  _
     ( T  T     +   T  T  ) / ( T  T  ) = ( -17.7559457 , 144.8169303 )
        1  4  ( - )  2  3        2  4
                                          ( 3.03347789 , 84.63484087 )

          _    _          _    _   _    _   _    _   _    _   _  _   _  _
29.  If   T  = T  , then  T  + T = T  + T , T  - T = T  - T , T  T = T  T  and
           1    2          1        2        1        2        1      2
     _     _    _     _        _     _
     T  /  T  = T  /  T , if   T < > 0 .
      1          2
       _   _      _    _        _  _        _  _
30.  ( T / T  ) ( T  / T  ) = ( T  T  ) / ( T  T  )
        1   2      3    4        1  3        2  4
       _    _        _    _        _  _        _  _
31.  ( T  / T  ) / ( T  / T  ) = ( T  T  ) / ( T  T  )
        1    2        3    4        1  4        2  3
       _    _          _    _      _
32.    T  / T   +  ( - T  / T  ) = 0
        1    2          1    2
        _    _      _    _      _         _    _        _
33.   ( T  / T  ) ( T  / T  ) = 1 , if  ( T  / T  ) < > 0
         1    2      2    1                2    1

       _  _   -1   _ -1 _ -1     _   -1      _ -1          _      _      _
34.  ( T  T  )   = T    T   , ( -T  )   = -( T    ) , if   T  and T  < > 0
        1  2        2    1        1           1             1      2
      _    _    _        _  _      _         _      _
35.   T  ( T  / T  ) = ( T  T  ) / T  , if   T  < > 0
       1    2    3        1  2      3         3

      _          _    _                     _  _       _       _       _      _
36.   T    +   ( T  / T  ) is in gen. < > ( T  T   +   T  ) /  T  ,if  T  < > 0
       1 ( - )    2    3                     1  3 ( - )  2       3       3

      (counter example)
      _                  _                    _
      T  = ( 15 , 32 ) , T  = ( 25 , 60 ) ja  T  = ( 32 , 60 )
       1                  2                    3
      _    _
      T  / T  = ( 353 , 85.94366927 )
       2    3
      _  _
      T  T   = ( 47 , 22.34021442 )
       1  3
      _          _    _
      T    +   ( T  / T  ) = ( 0.594677674 , 117.0912043 )  (check direction)
       1 ( - )    2    3
                             ( -11.58022643 , 56.62678908 ), q = 166.2486907,
                                                             w = 26.58022643
      _  _        _
      T  T    +   T  = ( 31.66542675 , 81.45083802 )
       1  3 ( - )  2
                       ( 77.414227 , 39.85373444 ) , q = 167.1619383,
                                                     w = 30.41422699
        _  _        _      _
      ( T  T    +   T  ) / T  = ( -0.33457325 , 77.77982094 )
         1  3 ( - )  2      3
                                ( 45.414227 , 38.05751302 )

        _    _      _      _      _  _      _    _    _
37.   ( T  / T  ) / T  = ( T  / ( T  T  ) , T  / 1  = T
         1    2      3      1      2  3      1         1
         _    _          _      _    _        _
38.   -( T  / T  ) = ( - T  ) / T  = T  / ( - T  )
          1    2          1      2    1        2
          _          _        _    _
39.   ( - T  ) / ( - T  )  =  T  / T
           1          2        1    2

40. In general: Product is not injective mapping.

41. Four operations of complex numbers comes in special case of plane surface,
    Then all axioms of complex numbers holds.


    CASE 1's  INTERPRETATION:

    On the surface angle sum is generally different than pii radians.

    CASE 2's INTERPRETATION:

    On the surface there does not generally exist similarity study of plane
    surface.




    ASSOCIATOR:


    Associator measures deviation from associative axiom.
    It is defined in the following way:
     _          _       _    _            _    _      _
     A    =  (  T  +  ( T  + T  ) ) - ( ( T  + T  ) + T  )
      123        1       2    3            1    2      3

    This can be used instead of associative axiom.


    DISTRIBUTOR:

    Distributor measures deviation from distributive axiom.
    It is defined in the following way:
     _ 1    _   _    _        _  _    _  _
     D    = T ( T  + T  ) - ( T  T  + T  T  )
       23    1   2    3        1  2    1  3

    This can be used instead of distributive axiom.

    Rem. Associator and distributor could possibly be used to describe
    curvature of the surface.
                                                        _ *
    COMPLEMENT NUMBER OF GENERAL ANGLE COMPLEX NUMBER   T


    There exists unigue complement general angle complex number for every
    general angle, which is defined as general angle complex number.
    It starts from the apex point of the preliminary general angle complex
    number, goes same direction as preliminary one along the sphere surface,
    and ends to the starting point of the preliminary one.
       _     _ *
       T  +  T   length is equal to  2*pii*R  and
                         _ *              _
       rotation angle of T    is same as  T has.


    COMPLEMENT PARALLELOGRAM

    Complement parallelogram is the external part of the preliminary
    parallelogram on the sphere. Its sum diagonal is extended part of
    preliminary sum diagonal (to the length 2*pii*R). It's subtraction
    diagonal is extended part of preliminary subtraction diagonal
    (to the length 2*pii*R)



 ALGEBRA OF GENERAL PSEUDO ANGLE COMPLEX NUMBERS


 Replace R by iS in formulas of general angle complex numbers, then you
 get corresponding formulas of this algebra, when you use following relations:
 sin( iS ) = i sinh S , cos( iS )= coshS ja tan( iS ) = i tanh S ,
 where i is imaginary unit, (Peirce, Foster 1956, page 81).
 Remember pseudo-sphere is investigated in frame where we have coordinates:
 (ct,x,y) and so called Minkowski metric ( Dubrovin,Fomenko, Novikov, Part 1,
 1984, pages 20-23, 50-59 and 90-95).
  This algebra is useful because there is no limitations about lengths of
 general pseudo angle complex numbers.



  EQUATIONS OF SPHERE SURFACE AND PSEUDO-SPHERE SURFACE


                             3
    Components of sphere in R :

    f ( u, v ) = R sin v sin u
     1

    f ( u, v ) = R cos v sin u
     2

    f ( u, v ) = R cos u
     3

    R => 0, 0 <= v < 2 Pii , 0 <= u <= Pii

    ( Psi <--> v , Theta <--> u )

    Components of pseudo-sphere: ( f  corresponds to base vector e  ):
                                    0                             0

    f ( u, v ) = S cosh u         = ct
     0

    f ( u, v ) = S cos v sinh u   = x
     1

    f ( u, v ) = S sin v sinh u   = y
     2

    -infinite < S < +infinite, 0 <= v < 2*pii , 0 <= u < +infinite

    ( Psi <--> v , Theta <--> u )

    Second equation of pseudo-sphere:
     2 2    2    2    2
    c t  - x  - y  = S
    Minkowski metric has following form in case of pseudo-sphere:
      2      2     2       2      2       2
    dl  = - S  ( du  + sinh u  d v  ) + dS
    , which has on upper hyperboloid sheet (dS = 0, and S = const.)
    the following form:
      2      2     2       2      2
    dl  = - S  ( du  + sinh u  d v  )
    Absolute value of this is called Lobachevsky metrics, which has second
    form in so called Poincare model of Lobachevsky geometry:
      2      2     2      2      2   2   2
    dl  = 4 S  ( dU  +  dV )/ ( S - U - V  ),

            2    2    2
    where  U  + V  < S  .
    Second form of mertics of pseudo-sphere is:
      2    2  2     2     2
    dl  = c dt  - dx  - dy
   (Encyclopaedia of Mathematics 1988-1990, and
    Dubrovin Fomenko Novikov 1984, Part 1, pages 8 and 22 and 90-95)


    REFERENCES CITED:


 1.  Ayres F.,JR., 1954.
     Theory and Problems of Plane and Spherical Trigonometry.
     Schaum's Outline Series, McGraw-Hill book Company, New York,
     United States of America.

 2.  Dubrovin, B. A., Fomenko, A. T., Novikov, S. P., 1984.
     Modern Geometry- Methods and Applications.
     Part 1, 2, 3, Springer-Verlag, New York,
     United States of America.

 3.  Encyclopaedia of Mathematics, 1988 - 1990.
     Volumes 1 - 6 , ( 7 - 10  not yet published ),
     Reidel, Kluver Academic Publishers, Science and Technology Divisions,
     Netherlands.

 4.  Lipschutz Martin M., 1969.
     Theory and Problems of Differential Geometry.
     Schaum's Outline Series, McGraw-Hill Book Company, New York,
     United States of America.

 5.  Peirce, B. O., Foster Ronald M., 1957.
     A Short Table of Integrals.
     Fourth edition, Blaisdell Publishing Company, A Division of Ginn and
     Company, United States of America.

 6.  Rashevskii, P. K, 1953.
     Riemannian Geometry and Tensor Analysis.
     Gostekhizdat, Page 413, (in Russian).

 7.  Rodichev, V. I., 1961.
     Twisted Space and Nonlinear Field Equations.
     Soviet Physics JETP, Volume 13, Number 5, November 1961, Pages 1029 - 1031.


X-News: tnclus sci.math:9408
From: hporopudas@tnclus.tele.nokia.fi
Subject:Re: Surface Algebras (one open problem)
Date: Thu, 7 May 1992 14:19:53 GMT
Message-ID:<1992May7.161953.1@tnclus.tele.nokia.fi>


 Surface algebras:

 About the case: sum, subtraction, product and division of two directed
 geodesic lines, which starts from two different points on surface:


   While ago I posted here my papers concernig new surface algebras,
 I have not got enough valuable comments about it.
   I leave to be further investigated one important open question in my
 surface algebras:


   I must say that there are several possibilities to define sum, subtraction,
 product and division for directed geodesic lines (on surface in three
 dimensional Euclidean space) in case of directed geodesic lines, which starts
 from two different points. One way is what I used and it gave algebras
 defined. Other possibilities will lead us numerous completely different
 algebras.
   I have become little suspicious about that way I used, because of physical
 interpretation that I gave. It is the most important desired property of
 the surface algebra.
   Physical interpretation should be used as quiding principle when trying
 to define this case. One physical example which could help in this case is
 two different geodesic flows on surface, which starts from two different
 points on the surface. That how these flows combine gives sum definition to
 the surface algebras. (It could be worth to investigate if sum diagonal
 should start from some point on geodesic line joining these two points.)
   That how subtraction is formed remains here open.
 (Somehow parallelogram, which have sum and subtraction geodesics as diagonals
  and geodesic line joining end points of preliminary directed geodesic lines
  could be possible used.)
 Product and division is formed in starting point of sum geodesic.


( ABSTRACT:

          The paper is a generalizaton of the algebra of complex numbers.
          The concept of complex numbers is usually restricted to the
          plane surface. Plane restriction is removed and new numbers
          called curve complex numbers or directed geodesic lines are defined
          on the regular surfaces. This is done by defining a geodesic
          parallelogram on the regular surface so that it is an object which
          is closed quadrangle that has equal length opposite geodesics.
          In this way, addition and subtraction of these new numbers is
          defined as diagonal geodesics of the geodesic parallelogram.
          One problem was how to transfer the subtraction diagonal geodesic
          beginning from the same starting point. The solution to this
          problem was to use these new geodesic parallelograms on the surface
          so that the transferred directed geodesic line was pointing always
          away from the starting point.
          Multiplication was defined with respect to some reference
          direction as adding up rotation angles and multiplying the
          corresponding lengths. Division was defined with respect to the same
          reference direction as subtracting rotation angles and dividing
          the corresponding lengths.
          The resultant surface algebra has two characteristic features
          namely non-associativity in respect to addition, which is the result
          of the fact that in general on the regular surface the angle sum
          of the geodesic triangle is different from 3.14159...  and
          non-distributivity (means nonlinearity) which is the result of
          the fact that these geodesic triangles have no similarity study on
          the regular surface as they have on the plane.
          In general Riemannian space and Riemannian-Cartan space these
          algebras are called surface algebras.
          The following two special algebras are treated more closely:
           - on the sphere of radius R,
           - on the pseudosphere of radius S,
          On the sphere, new numbers are called angle complex numbers (R=1)
          and on the pseudosphere pseudo angle complex numbers (S=1).  )


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