Qualitative Description of the Hyperbolic Plane
The hyperbolic plane is qualitatively different from the Euclidean plane in a number of ways. Among these are the fact that the sum of the angles of a triangle is strictly less than π (radians). The difference between π and the sum is called the defect of the triangle and is proportional to the area of the triangle, so the areas of triangles are bounded above. It is true in the hyperbolic plane that the length of a circular arc and the area of a circular sector are both proportional to the angle of the arc. What is different is the way the proportionality depends on the radius of the circle. Another difference is in the structure of the group of isometries, which are mappings of the plane that preserve distance. Since tilings and lattices are tied so closely to the isometries of the plane, they are also very different from the Euclidean case. All of the distinctive properties can be traced to the hyperbolic parallel axiom, so we begin with the theory of parallelism, presented in roughly the form in which Gauss derived it.
KeywordsIdeal Point Euclidean Plane Hyperbolic Plane Polar Coordinate System Euclidean Case
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