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There are many more metric properties of hyperbolic space that differentiate it from Euclidean space. Some can be generalised to the setting of Gromov-hyperbolic spaces, which is a generalisation of the notion of negative curvature to general metric spaces using only the large-scale properties. A finer notion is that of a CAT(−1)-space.
The group SO + (1,n) is the full group of orientation-preserving isometries of the n-dimensional hyperbolic space. In more concrete terms, SO + (1,n) can be split into n(n-1)/2 rotations (formed with a regular Euclidean rotation matrix in the lower-right block) and n hyperbolic translations, which take the form
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations) "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
In dimension 3, the fractional linear action of PGL(2, C) on the Riemann sphere is identified with the action on the conformal boundary of hyperbolic 3-space induced by the isomorphism O + (1, 3) ≅ PGL(2, C). This allows one to study isometries of hyperbolic 3-space by considering spectral properties of representative complex matrices.
Tessellations of hyperbolic 2-space are hyperbolic tilings. There are infinitely many regular tilings in H 2. As stated above, every positive integer pair {p,q} such that 1/p + 1/q < 1/2 gives a hyperbolic tiling. In fact, for the general Schwarz triangle (p, q, r) the same holds true for 1/p + 1/q + 1/r < 1.
The hyperbolastic rate equation of type II, denoted by H2, is defined as = (() ()),where is the hyperbolic tangent function, is the carrying capacity, and both and > jointly determine the growth rate.
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The metric of the model on the half-plane, { , >}, is: = + ()where s measures the length along a (possibly curved) line. The straight lines in the hyperbolic plane (geodesics for this metric tensor, i.e., curves which minimize the distance) are represented in this model by circular arcs perpendicular to the x-axis (half-circles whose centers are on the x-axis) and straight vertical rays ...