9: X-Y Coordinate System

The figure above shows 24 infinite lines which can be used to define a coordinate system in hyperbolic geometry. Point X is at the origin of this coordinate system. Let the horizontal and the vertical lines that intersect at the origin be the x-axis and y-axis. These axes dived the hyperbolic plane into four quadrants analogous to the four quadrants in the usual Cartesian coordinate system in Euclidean geometry. In the figure, each the axis is marked off by perpendicular lines that intersect the axis at intervals of 0.5 units. For example, the length of segment XA = AB = XS = ST = 0.5 units. In the Euclidean geometry, Cartesian coordinate system, the coordinates of any point in the first quadrant are defined to be the ordered pair, (x,y) where x is the perpendicular distance from the point to the x-axis, and y is the perpendicular distance from the point to the y-axis. Points in the second, third and forth quadrants are (-x,y), (-x,-y) and (x,-y) respectively. This same definition can be applied to the Hyperbolic plane so that every point in the plane has a unique set of coordinates. In spite of the fact that we use the same definition for coordinates in both geometeries, we cannot use the usual Euclidean method of locating points in the Hyperbolic plane. For example, in Euclidean geometry, to locate the point (1,1), we might first locate the perpendicular to the x-axis that is one unit from the origin, then locate the perpendicular to the y-axis that is one unit from the origin, and finally locate the intersection of these perpendiculars. This procedure, however, does not work in hyperbolic geometry. Notice that the perpendicular to the x-axis that is one unit from the origin (at point B), and the perpendicular to the y-axis that is one units from the origin (at point T), do not intersect! This might make it seem like the point (1,1) is undefined in hyperbolic geometry; however, the point (1,1) does exist, and it is located at point P. The length of the perpendicular from P to the x-axis is 1.0 units. Likewise, the length of the perpendicular from P to the y-axis is also 1.0 units. Yet the distance from the origin to the point where the perpendicular crosses the x-axis is only 0.7 units.

This coordinate system sets up a one-to-one correspondence between all of the points in the Hyperbolic plane and all ordered pairs (x,y) where x and y are real numbers.

What does the equation: y=x² look like in hyperbolic geometry?