# the hyperbolic plane

 Here, we introduce the Poincare disc model of hyperbolic geometry. Our description uses ideas from inversive geometry. We can get a similar model for euclidean geometry. The set is the open disc D:{z : |z| < 1} in the complex plane. Note. The boundary of D is the unit circle C: {z : |z| = 1}. The points of C do not belong to the geometry, but they play a role similar to the points at infinity in euclidean geometry. Definition A hyperbolic line (or h-line) is a subset of D of the form LnD, where L is an i-line orthogonal to C. An i-line L is either an extended line or a circle. Observe that an extended line L is orthogonal to C if and only if 0εL. In such a case, the h-line LnD is a diameter of D. The concept of orthogonal circles is less familiar. We have the following facts: If circle C (centre O) is orthogonal to circle L (centre P), then O lies outside L, and P lies outside C. If P is a point outside C, then there is a unique circle with centre P orthogonal to C. proofs of these facts The figure illustrates the h-line obtained from the point P. The complete orthogonal circle is shown, h-line is shown in red. You may move the point P to see how the h-line varies. If you drag P inside C, there is no circle L, and hence no h-line.