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In hyperbolic geometry, angle of parallelism () is the angle at the non-right angle vertex of a right hyperbolic triangle having two asymptotic parallel sides. The angle depends on the segment length a between the right angle and the vertex of the angle of parallelism. Given a point not on a line, drop a perpendicular to the line from the point.
In the Beltrami-Klein model of the hyperbolic geometry: two ultraparallel lines correspond to two non-intersecting chords. The poles of these two lines are the respective intersections of the tangent lines to the boundary circle at the endpoints of the chords. Lines perpendicular to line l are modeled by chords whose extension passes through ...
Using a hyperbolic ruler, construct a line c such that c is perpendicular to b and parallel to a. As a result, c is also parallel to a', making c the common parallel to lines a and a'. [3] Case 2: a and a' are parallel to each other. Using a hyperbolic ruler, construct AI' such that AI' is parallel to a' and perpendicular to a.
Compared to Euclidean geometry, hyperbolic geometry presents many difficulties for a coordinate system: the angle sum of a quadrilateral is always less than 360°; there are no equidistant lines, so a proper rectangle would need to be enclosed by two lines and two hypercycles; parallel-transporting a line segment around a quadrilateral causes ...
English: In the Poincaré disc model of the hyperbolic plane, lines are represented by circular arcs orthogonal to the boundary of the closure of the disc. The thin black lines meet at a common point and do not intersect the thick blue line, illustrating that in the hyperbolic plane there are infinitely many lines parallel to a given line passing through the same point.
The two lines through a given point P and limiting parallel to line R.. In neutral or absolute geometry, and in hyperbolic geometry, there may be many lines parallel to a given line through a point not on line ; however, in the plane, two parallels may be closer to than all others (one in each direction of ).
For example, in thermodynamics the isothermal process explicitly follows the hyperbolic path and work can be interpreted as a hyperbolic angle change. Similarly, a given mass M of gas with changing volume will have variable density δ = M / V , and the ideal gas law may be written P = k T δ so that an isobaric process traces a hyperbola in the ...
Given any line L and point P not on L, there are at least two distinct lines passing through P that do not intersect L. It is then a theorem that there are infinitely many such lines through P. This axiom still does not uniquely characterize the hyperbolic plane up to isometry; there is an extra constant, the curvature K < 0, that must be ...
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