Here, we introduce the Poincare disc model of hyperbolic geometry. Our description uses ideas from inversive geometry.
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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
An i-line L is either an extended line or a circle. Observe that
The concept of orthogonal circles is less familiar. We have the following facts:
The figure illustrates the h-line obtained from the point P.
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