Geometry | Part IB, 2003

Describe the geodesics (that is, hyperbolic straight lines) in either the disc model or the half-plane model of the hyperbolic plane. Explain what is meant by the statements that two hyperbolic lines are parallel, and that they are ultraparallel.

Show that two hyperbolic lines ll and ll^{\prime} have a unique common perpendicular if and only if they are ultraparallel.

[You may assume standard results about the group of isometries of whichever model of the hyperbolic plane you use.]

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