next up previous contents index
Next: Index Up: Hyperbolic Analytic Geometry Previous: More on Quadrilaterals

Coordinate Geometry in the Hyperbolic Plane

In the hyperbolic plane choose a point O for the origin and choose two perpendicular lines through O--tex2html_wrap_inline20607 and tex2html_wrap_inline20609. 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 tex2html_wrap_inline20617 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 tex2html_wrap_inline20633 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 tex2html_wrap_inline20643 be the hyperbolic distance from O to P and let tex2html_wrap_inline11280 be a real number so that tex2html_wrap_inline20651. Then

We also set

The ordered pair tex2html_wrap_inline20653 is called a frame with axes tex2html_wrap_inline20607 and tex2html_wrap_inline20609. With respect to this frame, we say the point P has

If a point has Beltrami coordinates (x,y) and tex2html_wrap_inline20673, 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 tex2html_wrap_inline20677, u=a is the equation of a line;
  2. for tex2html_wrap_inline20677, w=a is the equation of a hypercycle;
  3. tex2html_wrap_inline20685 is an equation of the line in the first quadrant that is horoparallel to both axes.
  4. tex2html_wrap_inline20687 is an equation of the horocycle with radius tex2html_wrap_inline20689.
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 tex2html_wrap_inline20695. 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 tex2html_wrap_inline20709. It then can be shown that (u,v) are the axial coordinates of a point if and only if tex2html_wrap_inline20713.

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 tex2html_wrap_inline20717.
  2. If the point tex2html_wrap_inline20719 has Beltrami coordinates tex2html_wrap_inline20721 and point tex2html_wrap_inline20723 has Beltrami coordinates tex2html_wrap_inline20725, then the distance tex2html_wrap_inline20727 is given by the following formulæ:
  3. Ax+By+C=0 is an equation of a line in Beltrami coordinates if and only if tex2html_wrap_inline20731, and every line has such an equation.
  4. Given an angle tex2html_wrap_inline20733 and given that the Beltrami coordinates of P are tex2html_wrap_inline20721, of Q are tex2html_wrap_inline20725, and of R are tex2html_wrap_inline20745, 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 tex2html_wrap_inline11280 is the angle formed by their intersection, then
    In particular the lines are perpendicular if and only if AD+BE=CF.
  6. If tex2html_wrap_inline20721 and tex2html_wrap_inline20725 are the Beltrami coordinates of two distinct points, let tex2html_wrap_inline20759 and tex2html_wrap_inline20761. 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 tex2html_wrap_inline20763 and tex2html_wrap_inline20765 are equations of lines in Beltrami coordinates and if tex2html_wrap_inline20767, 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 tex2html_wrap_inline20769 and c>0.
      2. The cycle is a horocycle if and only if tex2html_wrap_inline20773 and c>0.
      3. The cycle is a hypercycle if and only if tex2html_wrap_inline20777.

In Poincaré coordinates  (p,q)
is an equation of a line if and only if tex2html_wrap_inline20731, and every line has such an equation.

next up previous contents index
Next: Index Up: Hyperbolic Analytic Geometry Previous: More on Quadrilaterals