Coordinate Geometry in the Hyperbolic Plane

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

• axial coordinates (u,v), 
• polar coordinates , 
• Lobachevsky coordinates (u,w), 
• Beltrami coordinates (x,y), 
• Weierstrauss coordinates (T,X,Y). 

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

1. for , u=a is the equation of a line;
2. for , w=a is the equation of a hypercycle;
3. is an equation of the line in the first quadrant that is horoparallel to both axes.
4. is an equation of the horocycle with radius .

Thus, a line does not have a linear equation in Lobachevsky coordinates, and a linear equation does not necessarily describe a line.

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

1. Every point has a unique ordered pair of Beltrami coordinates , and (x,y) is an ordered pair of Beltrami coordinates if and only if .
2. If the point has Beltrami coordinates and point has Beltrami coordinates , then the distance is given by the following formulæ:
   
   
3. Ax+By+C=0 is an equation of a line in Beltrami coordinates if and only if , and every line has such an equation.
4. Given an angle and given that the Beltrami coordinates of P are , of Q are , and of R are , then the cosine of this angle is given by
   
   
5. If Ax+By+C=0 and Dx+Ey+F=0 are equations of two intersecting line in Beltrami coordinates and is the angle formed by their intersection, then
   
   In particular the lines are perpendicular if and only if AD+BE=CF.
6. If and are the Beltrami coordinates of two distinct points, let and . Then the midpoint of the segment joining the two points has Beltrami coordinates
   
   and the perpendicular bisector of the two points has an equation
   
   
7. If and are equations of lines in Beltrami coordinates and if , then the two lines are hyperparallel.
8. Every cycle has an equation in Beltrami coordinates that is of the form
   
   
   1. The cycle is a circle if and only if and c>0.
   2. The cycle is a horocycle if and only if and c>0.
   3. The cycle is a hypercycle if and only if .

In Poincaré coordinates  (p,q)

is an equation of a line if and only if , and every line has such an equation.