In the hyperbolic plane choose a point O for the origin and choose two
perpendicular lines through O-- and . In our
models--both the Klein and Poincaré--we will use the Euclidean center of
our defining circle for this point O. We need to fix coordinate systems on
each of these two perpendicular lines. By this we need to choose a positive and
a negative direction on each line and a unit segment for each. There are other
coordinate systems that can be used, but this is standard. We will call these
the u-axis and the v-axis. For any point let U and V be
the feet of P on these axes, and let u and v be the respective
coordinates of U and V. Then the quadrilateral is a Lambert
quadrilateral. If we label the length of UP as w and that of VP as z,
then by the Corollary to Theorem 22.1 we have
Let be the hyperbolic distance from O to P and let be
a real number so that . Then
We also set
The ordered pair is called a frame with axes and . With respect to this frame, we say the point P has
If a point has Beltrami coordinates (x,y) and , put
then (p,q) are the Poincaré coordinates of the
point.
Every point has a unique ordered pair of Lobachevsky coordinates , and, conversely, every ordered pair of real numbers is tha pair of Lobachevsky coordinates for some unique point. In Lobachevsky coordinates
Every point has a unique ordered pair of axial coordinates . However, not every ordered pair of real numbers is a pair of axial coordinates. Let U and V be points on the axes with . Now the perpendiculars at U and V do not have to intersect. It is easy to see that they might be horoparallel or hyperparallel, especially by looking in the Poincaré model. If the two lines are limiting parallel (horoparallel) then that would make the segments OU and OV complementary segments. It can be shown then that these perpendiculars to the axes at U and V will intersect if and only if . It then can be shown that (u,v) are the axial coordinates of a point if and only if .
Lemma 21.1: With respect to a given frame
In Poincaré coordinates (p,q)
is an equation of a line if and only if , and every line has such
an equation.