_ξ_ = _η_ = _ζ_ = 0
from which equation 1 gives _τ_ = _R_, then,
( _s_₀ ) _σ_
sin ( ---- ) = -----
( _R_ ) _R_
in which
____________________
_σ_ = √(_x_² + _y_² + _z_²)
or,
( _σ_ ) ( _σ_ )
_s_₀ = _R_ . arc sin( --- ) = _R_ . arc tang( --- ) (5.)
( _R_ ) ( _t_ )
In this, _s_₀ is the distance of the point _x_, _y_, _z_, measured from
the centre of the co-ordinates.
If now we suppose the point _x_, _y_, _z_, of spherical space, to be
projected in a point of plane space whose co-ordinates are respectively
( _Rx_ ) ( _Ry_ ) ( _Rz_ )
χ = ( ---- ) ϒ = ( ---- ) ζ = ( ---- )
( _t_ ) ( _t_ ) ( _t_ )
_R_²_σ_²
χ² + ϒ² + ζ² = _r_² = ---------
_t_²
then in the plane space the equations 3, which belong to the
straightest lines of spherical space, are equations of the straight
line. Hence the shortest lines of spherical space are represented in
the system of χ, ϒ, ζ, by straight lines. For very small values of
_x_, _y_, _z_, _t_ = _R_, and χ = _x_, ϒ = _y_, ζ = _z_
Immediately about the centre of the co-ordinates, the measurements of
both spaces coincide. On the other hand, we have for the distances from
the centre
( _r_ )
_s_₀ = _R_ . arc tang( ± ---- ) (6.)
( _R_ )
In this, _r_ may be infinite; but every point of plane space must be
the projection of two points of the sphere, one for which _s_₀ <
½_R_π, one for which _s_₀ > ½_R_π. The extension in the direction of
_r_ is then
_ds_₀ _R_²
----- = -------------
_dr_ _R_² + _r_²
In order to obtain corresponding expressions for pseudospherical space,
let _R_ and _t_ be imaginary; that is, _R_ = ℛ_i_, and _t_ = τ_i_.
Equation 6 gives then
_s_₀ _r_
tang ------ = ± ------
_i_ℛ _i_ℛ
from which, eliminating the imaginary form, we get
ℛ + _r_
_s_₀ = ½ℛ log. nat. ---------
ℛ - _r_
Here _s_₀ has real values only as long as _r_ = R; for _r_ = ℛ the
distance _s_₀ in pseudospherical space is infinite. The image in plane
space is, on the contrary, contained in the sphere of radius _R_,
and every point of this sphere forms only one point of the infinite
pseudospherical space. The extension in the direction of _r_ is
_ds_₀ ℛ²
----- = -----------
_dr_ ℛ² - _r_²
For linear elements, on the contrary, whose direction is at right
angles to _r_, and for which _t_ is unchanged, we have in both cases
_____________________
√_dx_² + _dy_² + _dz_² _t_ τ _σ_
---------------------- = ----- = ---- = -----
_____________________ _R_ ℛ _r_
√_d_χ² + _d_ϒ² + _d_ζ²
__________________
√_x_² + _y_² + _z_²
= -------------------
_____________
√χ² + ϒ² + ζ²