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Hyperbolic motion
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Hyperbolic motion
In geometry, hyperbolic motions are isometric automorphisms of a hyperbolic space. Under composition of mappings, the hyperbolic motions form a continuous group. This group is said to characterize the hyperbolic space. Such an approach to geometry was cultivated by Felix Klein in his Erlangen program. The idea of reducing geometry to its characteristic group was developed particularly by Mario Pieri in his reduction of the primitive notions of geometry to merely point and motion.
Hyperbolic motions are often taken from inversive geometry: these are mappings composed of reflections in a line or a circle (or in a hyperplane or a hypersphere for hyperbolic spaces of more than two dimensions). To distinguish the hyperbolic motions, a particular line or circle is taken as the absolute. The proviso is that the absolute must be an invariant set of all hyperbolic motions. The absolute divides the plane into two connected components, and hyperbolic motions must not permute these components.
One of the most prevalent contexts for inversive geometry and hyperbolic motions is in the study of mappings of the complex plane by Möbius transformations. Textbooks on complex functions often mention two common models of hyperbolic geometry: the Poincaré half-plane model where the absolute is the real line on the complex plane, and the Poincaré disk model where the absolute is the unit circle in the complex plane. Hyperbolic motions can also be described on the hyperboloid model of hyperbolic geometry.
This article exhibits these examples of the use of hyperbolic motions: the extension of the metric to the half-plane and the unit disk.
Every motion (transformation or isometry) of the hyperbolic plane to itself can be realized as the composition of at most three reflections. In n-dimensional hyperbolic space, up to n+1 reflections might be required. (These are also true for Euclidean and spherical geometries, but the classification below is different.)
All the isometries of the hyperbolic plane can be classified into these classes:
The points of the Poincaré half-plane model HP are given in Cartesian coordinates as {(x,y): y > 0} or in polar coordinates as {(r cos a, r sin a): 0 < a < π, r > 0 }. The hyperbolic motions will be taken to be a composition of three fundamental hyperbolic motions. Let p = (x,y) or p = (r cos a, r sin a), p ∈ HP.
The fundamental motions are:
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Hyperbolic motion
In geometry, hyperbolic motions are isometric automorphisms of a hyperbolic space. Under composition of mappings, the hyperbolic motions form a continuous group. This group is said to characterize the hyperbolic space. Such an approach to geometry was cultivated by Felix Klein in his Erlangen program. The idea of reducing geometry to its characteristic group was developed particularly by Mario Pieri in his reduction of the primitive notions of geometry to merely point and motion.
Hyperbolic motions are often taken from inversive geometry: these are mappings composed of reflections in a line or a circle (or in a hyperplane or a hypersphere for hyperbolic spaces of more than two dimensions). To distinguish the hyperbolic motions, a particular line or circle is taken as the absolute. The proviso is that the absolute must be an invariant set of all hyperbolic motions. The absolute divides the plane into two connected components, and hyperbolic motions must not permute these components.
One of the most prevalent contexts for inversive geometry and hyperbolic motions is in the study of mappings of the complex plane by Möbius transformations. Textbooks on complex functions often mention two common models of hyperbolic geometry: the Poincaré half-plane model where the absolute is the real line on the complex plane, and the Poincaré disk model where the absolute is the unit circle in the complex plane. Hyperbolic motions can also be described on the hyperboloid model of hyperbolic geometry.
This article exhibits these examples of the use of hyperbolic motions: the extension of the metric to the half-plane and the unit disk.
Every motion (transformation or isometry) of the hyperbolic plane to itself can be realized as the composition of at most three reflections. In n-dimensional hyperbolic space, up to n+1 reflections might be required. (These are also true for Euclidean and spherical geometries, but the classification below is different.)
All the isometries of the hyperbolic plane can be classified into these classes:
The points of the Poincaré half-plane model HP are given in Cartesian coordinates as {(x,y): y > 0} or in polar coordinates as {(r cos a, r sin a): 0 < a < π, r > 0 }. The hyperbolic motions will be taken to be a composition of three fundamental hyperbolic motions. Let p = (x,y) or p = (r cos a, r sin a), p ∈ HP.
The fundamental motions are: