Hyperbolic plane12/3/2023 ![]() ![]() The ultraparallel theorem states that there is a unique line in the hyperbolic plane that is perpendicular to each of a given pair of ultraparallel lines. ![]() The differences between these types of lines can also be looked at in the following way: the distance between asymptotic lines run to zero in one direction and grows without bound in the other the distance between ultraparallel lines increases in both directions. Thus we have this modified form of the parallel postulate: In hyperbolic geometry, given any line l, and point P not on l, there are exactly two lines through P which are asymptotic to l, and infinitely many lines through P ultraparallel to l. Notice that since there are an infinite number of possible angles between θ and 90 degrees, and each one will determine two lines through P and disjointly parallel to l, there exist an infinite number of ultraparallel lines. All other lines through P not intersecting l, with angles greater than θ with PB, are called ultraparallel (or disjointly parallel) to l. x and y are the only two lines asymptotic to l through P. Symmetrically, the line y that forms the same angle θ between PB and itself but clockwise from PB will also be asymptotic. This is called an asymptotic line in hyperbolic geometry. Consider the line x through P such that x does not intersect l, and the angle θ between PB and x counterclockwise from PB is as small as possible i.e., any smaller angle will force the line to intersect l. Let B be the point on l such that the line PB is perpendicular to l. In this article, the two limiting lines are called asymptotic and lines that have a common perpendicular are called ultraparallel the simple word parallel may apply to both.Īn interesting property of hyperbolic geometry follows from the occurrence of more than one parallel line through a point P: there are two classes of non-intersecting lines. These models prove that the parallel postulate is independent of the other postulates of Euclid.īecause there is no hyperbolic analogue to Euclidean parallel lines, the hyperbolic use of parallel and related terms varies among writers. Models have been constructed within Euclidean geometry that obey the axioms of hyperbolic geometry. ![]() As they do not intersect l, the parallel postulate is false. In hyperbolic geometry there are at least two such lines through P. The parallel postulate in Euclidean geometry says that in two dimensional space, for any given line l and point P not on l, there is exactly one line through P that does not intersect l. All glide reflections with the same translation length are conjugate to one another.In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced. Just like with hyperbolic isometries, a glide reflection has exactly two fixed points, namely the endpoints at infinity of the geodesic corresponding to the reflection and translation.Īlso like hyperbolic isometries, a glide reflection is determined by (1) its two fixed points, or equivalently the geodesic it fixes, (2) a direction, or orientation of the geodesic it fixes, and (3) a positive translation length. This is the sense in which one could classify orientation-reversing isometries with just one type.) (If we were to allow translations of length zero, then a reflection would be a type of glide reflection. Glide reflectionĪ glide reflection is an isometry that results from composing a reflection with a non-trivial translation (aka hyperbolic isometry) along the geodesic corresponding to the reflection. A reflection is uniquely determined by its geodesic, and every reflection is conjugate to every other one by an orientation-preserving isometry (so there is just one conjugacy class). Throughout this article we use \(H\) to denote the hyperbolic plane and \(\overline\) for a reflection is exactly the set of points making up the geodesic, including the geodesic's endpoints at infinity. The first row uses the Klein or projective model, and the second row the Poincaré disk model. This figure shows an animation of the three types of orientation-preserving isometries of the hyperbolic plane (from left to right): hyperbolic, elliptic, and parabolic. We discuss orientation-preserving isometries first after introducing some preliminaries. The isometries of the hyperbolic plane form a group under composition.Īn isometry of the hyperbolic plane can be either orientation-preserving or orientation-reversing. An isometry of the hyperbolic plane is a mapping of the hyperbolic plane to itself that preserves the underlying hyperbolic geometry (e.g. ![]()
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