From: hporopudas@tnclus.tele.nokia.fi
Newsgroups: sci.math
Subject: Surface Algebras (SUMMARY)
Date: 17 Apr 92 10:47:23 GMT
Organization: Nokia Telecommunications.


 SUMMARY OF SURFACE ALGEBRAS:



   While ago I posted here some papers which treated new surface algebras and
 I would like to put these papers here together mainly as it stands. I will
 make summary of possible comments later if I get them enough.
   These papers are the following:
     Message-ID:<1991Dec16.103041.1@tnclus.tele.nokia.fi>
     Message-ID:<1991Dec20.150729.1@tnclus.tele.nokia.fi>
     Message-ID:<1992Jan6.192226.1@tnclus.tele.nokia.fi>
     Message-ID:<1992Jan21.203922.1@tnclus.tele.nokia.fi>
     Message-ID:<1992Feb24.143618.1@tnclus.tele.nokia.fi>
   One purpose of surface algebras is try to write the theory of general
 relativity and some related theories simpler and more amateur understandable
 form. I don't yet know will this be successfull but anyhow I or perhaps
 somebody else will try it in due time.
   In mathematics one point of surface algebras is the definition of the
 "parallelogram" on curved surface. This gives different point of view to the
 standard "parallel translation" in differential geometry.
   I would also like to remind you that the algebra of complex numbers is
 here only a special case (a plane surface, which is achieved for example when
 R or S increases without limit).
   Lastly I would like to say that there is much more to be investigated
 in these new algebras that it little I have done here with my
 "experimental calculations".
   I recommend to use the following post address if you comment these
 surface algebras: Hannu Poropudas, Vesaisentie 9E, 90900 Kiiminki,
 Suomi-Finland. Or my present E-mail address: hporopudas@tnclus.tele.nokia.fi
 because I dont't always read postings in sci.math newsgroup.



 Oulu  -  Finland  15.4.1992   Hannu K. J. Poropudas

 Below this summary:


          SURFACE ALGEBRAS BASED ON DIFFERENTIAL GEOMETRY

               Hannu, K. J. Poropudas
               Vesaisentie 9E, 90900 Kiiminki, Finland.

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).


  GENERAL PART

  SURFACE ALGEBRAS BASED ON DIFFERENTIAL GEOMETRY:
  ALGEBRA OF CURVE COMPLEX NUMBERS


                                    3
  Let a surface in Euclidean space R  be represented on different domains
  in the following form:
  _   _                     _              _              _
  x = x ( u,v ) = f ( u,v ) e  + f ( u,v ) e  + f ( u,v ) e
                   1         1    2         2    3         3
                     _                      3
  , on which domains x belongs at least in C  (all directional derivatives
  exists and are continuous at least to third order) and
  _    _      _                             _
  x  x x  < > 0  (normal to the surface < > 0  on whole domain), and
   u    v
  _
  x  is bijective and continuous   f  , f  and f   are real valued functions
                                    1    2      3
                    _    _      _
  of two variables  e  , e  ja  e  are three orthonormal base vectors
                     1    2      3
    _   _       _       3
  ( i , j  and  k ) in R .

  (Concepts: pages 1-10, 21-31, 43-52, 61-71, 80-91, 102-111, 121-135,
   150-160, 171- 189, 201-215, 227-246, 263-265. Lipschutz 1969).

         _   _         _                                        2
  Curve  x = x ( s ) = x ( u(s),v(s) ), which belongs in class C  is
  geodesic if and only if

   2       2    1               2       1
  d u / d s + A    ( d u / d s )  + 2 A    ( d u / d s ) ( d v / d s ) +
                11                      12

                1               2
            + A    ( d v / d s )   = 0
                22

   2       2    2               2       2
  d v / d s + A    ( d u / d s )  + 2 A    ( d u / d s ) ( d v / d s ) +
                11                      12

                2               2
            + A    ( d v / d s )   = 0
                22

  u( 0 ) = u   ,  v( 0 ) = v   ,
            0               0

  d u( 0 ) / d s = ( d u / d s )   , d v( 0 ) / d s = ( d v / d s )   ,
                                0                                  0
           _         2                  2
  (abs ( d x / d s ))  = E ( d u / d s )  + 2 F ( d u / d s )  ( d v / d s )  +
                                        0                    0              0

                                          2
                         + G ( d v / d s )   =  1
                                          0
  (The last condition for geodesic line does not restrict its orientation
  (see pages 234-235 and 251. Lipschutz 1969).)

  Maximum value of arc length s of the geodesic line depends on the surface
  under study. Useful algebra is achieved when s is allowed to change between
  limits 0 and +infinite. For example on surface of periodic sphere multiples
  of 2R*3,1415927... radians must be subtracted away from arc length s of
  geodesic line under study. By this way achieved directed geodesic line is
  said to be equivalent with the primary one.

    k
  A     are so called Christoffel symbols of second kind,
    ij                          1
        which belongs in class C   .

    1                                           2
  A    = ( G E  - 2 F F  + F E  ) / ( 2( E G - F  ))
    11        u        u      v

    1                                  2
  A    = ( G E  - F G  ) / ( 2( E G - F  ))
    12        v      u

    1                                           2
  A    = ( 2 G F  - G G  - F G  ) / ( 2( E G - F  ))
    22          v      u      v

    2                                           2
  A    = ( 2 E F  - E E  + F E  ) / ( 2( E G - F  ))
    11          u      v      u

    2                                  2
  A    = ( E G  - F E  ) / ( 2( E G - F  ))
    12        u      v

    2                                           2
  A    = ( E G  - 2 F F  + F G  ) / ( 2( E G - F  ))   , where
    22        v        v      u
             _    _        _    _        _    _
         E = x  . x  , F = x  . x  , G = x  . x
              u    u        u    v        u    v

  E, F and G are functions in so called first fundamental form of the surface,
                          2
  which belongs in class C .

  (Theorm 11.8, page 234, proof on page 251, problem 11.14. Lipschutz 1969).
  _     _   _         _   _
  N = ( x x x ) / abs(x x x )   is unit normal of the surface.
         u   v         u   v
        _    _                _   _    _   _            _   _
  L = - x  . N  , M = -(1/2)( x . N  + x . N )  , N = - x . N
         u    u                u   v    v   u            v   v
        _    _        _    _         _    _
  L = - x  . N  , M = x  . N  ,  N = x  . N
         uu            uv             vv
  are functions in so called second fundamental form of the surface, which
                    1
  belongs in class C .

                     2
  Additionally EG - F  > 0, E > 0 and G > 0, and E, F, G, L, M, N satisfies so
  called compatibility conditions:
                  1         2      1           2
    L  - M  = L A    + M( A    - A     ) - N A
     v    u       1 2       1 2    1 1         1 1

                  1         2      1           2
    M  - N  = L A    + M( A    - A     ) - N A
     v    u       2 2       2 2    1 2         1 2

  and
          2         2          2          1     2      1     2
    LN - M  = F( (A     )  - (A     ) + A     A    - A     A      ) +
                    2 2  u      1 2  v    2 2   1 1    1 2   1 2

                    1           1         1     1      1     2
            + E( (A     )  - (A     ) + A     A    + A     A      ) +
                    2 2  u      1 2  v    2 2   1 1    1 2   2 2

                                          1     2      1     1
                                   - E( A     A    + A     A     )
                                          2 2   1 2    1 2   1 2

  These conditions carantees that there exists regular parametric repsentation
  for the surface:
  _   _                                     3
  x = x ( u , v ) , which belongs in class C  .

  (Existence theorem of surfaces, page 264. Lipschuts 1969).

  (Above "abs" means ordinary euclidic length of vector and "." is ordinary
   skalar product of vectors. Subscripts u and v means partial derivation
   with respect these parameters).


  PARALLELOGRAM ON THE SURFACE, SUM AND DIFFERENCE OF TWO DIRECTED GEODESIC
  LINES

                                 _
  When we take any regular point x  of this previous kind surface as an origin,
                                  0
  so we can define geodesic parallelogram, one corner starting from it, to be
  an object, which is closed figure on the surface formed by four geodesic
  lines, with the property that two opposite geodesics have egual arc lengths.

  For example the "north pole" can be chosen as an origin on sphere or on
  pseudosphere (see pseudosphere: Dubrovin Fomenko Novikov 1984).
                 _                                       _
  Let it be now  x( u(s),v(s) ); 0 <= s <=  s  , (marked T  )
                     1    1                  a            1
       _                                      _
  and  x( u(s),v(s) ); 0 <= s <= s  , (marked T  ) two through the point
           2    2                 b            2
  _                _                _
  x( u(0),v(0) ) = x( u(0),v(0) ) = x  ,
      1    1           2    2        0
  on the suface, passing directed geodesic lines, which are also called curve
  complex numbers, such that first has arc length s , and an other has arc
                                                   a
  length s .
          b
      _      _
  Let T  and T  be directed so that the following conditions hold:
       1      2
      _
  for T  :
       1

  u( 0 ) = u  , v( 0 ) = v  ,
   1        10   1        10

  d u( 0 ) / d s = ( d u / d s )  , d v( 0 ) / d s = ( d v / d s )   ,
     1                  1       1      1                  1       1

         _          2                  2
 (abs( d x  / d s ))  = E ( d u / d s )  + 2 F ( d u / d s ) ( d v / d s )  +
                               1       1            1       1     1       1

                                         2
                        + G ( d v / d s )    =  1
                                 1       1
                     _                                     _
 ( the definition of x  here: compare short hand notation  T  )
                                                            1
      _
  for T  :
       2

  u( 0 ) = u  , v( 0 ) = v  ,
   2        20   2        20

  d u( 0 ) / d s = ( d u / d s )  , d v( 0 ) / d s = ( d v / d s )   ,
     2                  2       2      2                  2       2

          _          2                  2
  (abs( d x  / d s ))  = E ( d u / d s )  + 2 F ( d u / d s ) ( d v / d s )  +
                                2       2            2       2     2       2

                                          2
                         + G ( d v / d s )    =  1
                                  2       2
                     _                                     _
 ( the definition of x  here: compare short hand notation  T  )
                                                            2
                        _                  _
 Now we draw, the point x( u(s ),v(s ) ) = y   as a center point, a geodesic
                            1 a   1 a       0
 circle, or in other words a track on the surface, which lies on constant
                                                                _
 distance s , measured along the surface, from the center point y  .
           b                                                     0
                      _                  _
 Similarly, the point x( u(s ),v(s ) ) = z  as a center point, a geodesic
                          2 b   2 b       0
 circle, or in other words a track on the surface, which lies on constant
                                                                _
 distance s , measured along the surface, from the center point z  .
           a                                                     0
                   _
 The cutting point w  of these tracks on the surface is the fourth corner
                    0
 point of the geodesic parallelogram.

 The second generally possible cutting point is the forth corner point of
 geodesic half-parallelogram (otherwise sayed: a closed figure on the surface,
 which has only one pair of equal arc length of opposite geodesic lines) is
 excluded from the following definitions.
                                                          _
 For regular surface a geodesic line starting from origin x  and ending to
                                                           0
 _
 w  exists, and this line is also unique.
  0
 (Theorem 11.9, page 235, Lipschutz 1969).

 We define that sum of two directed geodesic lines is the sum diagonal
                          _                                        _
 which starts from origin x  , goes along the surface, and ends to w  .
                           0                                        0


 PHYSICAL INTERPRETATION OF SUM OF TWO DIRECTED GEODESIC LINES

 The sum of two nonlinear directed geodesic lines (which are also called curve
 complex numbers) is interpreted as combining of two signals, two accelerations
 or two velocities, in nonlinear space.


 SUBTRACTION OF TWO DIRECTED GEODESIC LINES:

                                              _     _
 Subtraction of two directed geodesic lines:  T  -  T  is the directed geodesic
                                               2     1
                                   _
 line which starts from the point  x( u(s ),v(s ) ), goes along the suface,
                                       1 a   1 a
                       _
 and ends to the point x( u(s ),v(s ) )  .
                           2 b   2 b

                                              _     _
 Subtraction of two directed geodesic lines:  T  -  T  is the directed geodesic
                                               1     2
                                   _
 line which starts from the point  x( u(s ),v(s ) ), goes along the suface,
                                       2 b   2 b
                       _
 and ends to the point x( u(s ),v(s ) )  .
                           1 a   1 a

 This subtraction diagonal (dircted geodesic line) exists and it is also unique.
 (Theorem 11.9, page 235, Lipschutz 1969).

 The subtraction diagonal geodesic line is transferred beginning from the same
 starting point in the following way: We use these new geodesic parallelograms
 on the surface so that the transferred directed geodesic line is pointing
 always away from the starting point.

 This is the way also how to combine directed geodesic lines starting from
 two different starting point.

 PRODUCT OF TWO DIRECTED GEODESIC LINES:


 Let a common reference directed geodesic line of infinitedesimal length
 be such that it starts from the point
 _                _                 _
 x( u(0),v(0) ) = x ( u(0),v(0) ) = x ( u(0),v(0))
     1    1            2    2

 and holds following arbitarily chosen conditions for reference direction:

 d u(0) / d s = ( d u / d s )   and  d v(0) / d s = ( d v / d s ) .
                             r                                   r
 (compare x-axis in case of plane surface)
 In this way we choose certain fixed direction, from which we measure the
 rotation angle of directed geodesic line.
                                          _  _
 Product of two directed geodesic lines:  T  T  is directed geodesic line,
                                           1  2
 length of which is s  s  , and  direction of which is sum of directions
                     a  b
    _      _
 of T  and T  ,measured from the same reference direction, if such directed
     1      2
 geodesic line exists on the surface.



 QUOTIENT OF TWO DIRECTED GEODESIC LINES:

                                         _     _
 Quotient of two directed geodesic line: T  /  T  is directed geodesic line,
                                          1     2
 length of which is s  / s   , if  s  < > 0 and direction of which is
                     a    b         b
                             _      _
 difference of directions of T  and T  , measured from the same reference,
                              1      2
 if such directed geodesic line exists on the surface.

                                         _     _
 Quotient of two directed geodesic line: T  /  T  is directed geodesic line,
                                          2     1
 length of which is s  / s   , if  s  < > 0 and direction of which is
                     b    a         a
                             _      _
 difference of directions of T  and T  , measured from the same reference,
                              2      1
 if such directed geodesic line exists on the surface.

                                         _
 Rem. Multiplying directed geodesic line T  by a positive real number means
                                          1
 that the initial conditions are the same, only the length of this directed
 geodesic line is multiplied by this number. Multplying by a negative real
 number a means initial conditions:

        _
  for a T  :
         1

  u( 0 ) = u  , v( 0 ) = v  ,
   1        10   1        10

  d u( 0 ) / d s = - ( d u / d s )  , d v( 0 ) / d s = - ( d v / d s )   ,
     1                    1       1      1                    1       1

         _          2                  2
 (abs( d x  / d s ))  = E ( d u / d s )  + 2 F ( d u / d s ) ( d v / d s )  +
                               1       1            1       1     1       1

                                         2
                        + G ( d v / d s )    =  abs( a )
                                 1       1

 Both cases presupposes that corresponding directed geodesic line exists.



 TRANSFERRING REFERENCE DIRECTION TO DIFFERENT POINTS OF THE SURFACE


 At the beginning at the origin chosen reference direction can be transferred
 to the starting point of directed geodesic line, which starts from different
 point by the following way:
 We search an infinitedesimal small directed geodesic line on the surface,
 which has this reference direction as a tangent line, and then we move this
 with aid of parallelogram on the surface, to start from the same starting
 point as the other directed geodesic line. The desired reference direction is
 now tangent line to this transferred directed geodesic line.


 SUM, SUBTRACTION, MULTIPLICATION AND QUOTIENT OF DIRECTED GEODESIC LINES
 WHICH STARTS FROM DIFFERENT POINTS


 From two different point starting directed geodesic lines are combined so that
 we draw directed geodesic lines from starting point of the first to both ends
 of the other. Then we transfer this directed geodesic line by the same way
 as we did in case of definition of subtraction directed geodesic line, by
 using the definition of the parallelogram on the surface, to first one's
 starting point. Then we can perform operations of sum, subtraction,
 multiplication and quotient as above.



 REMARK ABOUT GENERALIZATIONS

 These definitions are easily generalized with aid some additional minimal
                          n
 measures to surfaces in R  and to those spaces, where directed geodesic
 lines are sensible, for example to those spaces where torsion tensor does
 not vanish. (Differential equations of geodesic lines in so. called
 Riemann-Cartan geometry are for example on page 413, Rashevskii 1953 and
 on page 1030, Rodichev 1961).



 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.

  I must leave this part away, because this paper is going to be to long.



  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:
                                                                       _
    Pseudopallopinnan komponentit ovat ( 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.