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= Euclidean Parallelism = ''t'' produces congruent angles.]] Given a line ''l'' and a point ''a'' not on that line, the following three definitions of a line ''m'' through ''a'' are equivalent definitions of parallel line in Euclidean Space : #Every point on ''m'' is located exactly the same minimum distance from the other line or plane ('equidistant lines'). #If ''m'' is extended to Infinity in either direction, it will just not intersect ''l''. #If the two lines are both intersected by a third line (a Transversal ) in the same plane, and the angles of intersection are equal, then the two lines are parallel. Parallel lines have to lie in the same plane. This is implied by the first definition, but is a prerequisite for the latter two. In other words, parallel lines must be located in the same plane, and parallel planes must be located in the same three-dimensional space. A parallel combination of a line and a plane may be located in the same three-dimensional space. Lines parallel to each other have the same gradient. Compare to Perpendicular . CONSTRUCTION The three definitions above lead to three different methods of construction of parallel lines. Another definition of parallel line that's often used is that two lines are parallel if they do not intersect. This definition is not so useful, as it does not prescribe a way of constructing parallel lines. = Extension to non-Euclidean geometry = In lines are not geodesics so the equidistant definition cannot be used. In the Euclidean plane, when two geodesics (straight lines) are intersected with the same angles by a transversal geodesic (see image), every (non-parallel) geodesic intersects them with the same angles. In both the hyperbolic and spherical plane, this is not the case. E.g. geodesics sharing a common perpendicular only do so at one point (hyperbolic space) or at two (antipodal) points (spherical space). In general geometry it is useful to distinguish the three definitions above as three different types of lines, respectively equidistant lines, '''parallel geodesics''' and '''geodesics sharing a common perpendicular'''. While in Euclidean geometry two geodesics can either intersect or be parallel, in general and in hyperbolic space in particular there are three possibilities. Two geodesics can be either: # intersecting: they intersect in a common point in the plane # parallel: they do not intersect in the plane, but do in the limit to infinity # ultra parallel: they do not even intersect in the limit to infinity In the literature ''ultra parallel'' geodesics are often called ''parallel''. ''Geodesics intersecting at infinity'' are then called ''limit geodesics''. SPHERICAL , the equivalent of a straight line in the spherical plane. Line ''c'' is equidistant to line ''a'' but is not a great circle. It is a parallel of latitude. Line ''b'' is another geodesic which intersects ''a'' in two antipodal points. They share two common perpendiculars (one shown in blue).]] In the spherical plane, all geodesics are Great Circles . Great circles divide the sphere in two equal Hemispheres and all great circles intersect each other. By the above definitions, there are no parallel geodesics to a given geodesic, all geodesics intersect. Equidistant lines on the sphere are called parallels of latitude in analog to Latitude lines on a globe. These lines are not geodesics. An object traveling along such a line has to Accelerate away from the geodesic it is equidistant to to avoid intersecting with it. When embedded in Euclidean space a Dimension higher, parallels of latitude can be generated by the intersection of the sphere with a plane parallel to a plane through the center. HYPERBOLIC In the hyperbolic plane, there are two lines through a given point that intersect a given line in the limit to infinity. While in Euclidean geometry a geodesic intersects its parallels in both directions in the limit to infinity, in hyperbolic geometry both directions have their own line of parallelism. When visualized on a plane a geodesic is said to have a left handed parallel and a '''right handed parallel''' through a given point. The angle the parallel lines make with the perpendicular from that point to the given line is called the '''angle of parallelism'''. The angle of parallelism depends on the distance of the point to the line with respect to the Curvature of the space. The angle is also present in the Euclidean case, there it is always 90° so the left and right handed parallels Coincide . The parallel lines divide the set of geodesics through the point in two sets: '''intersecting geodesics''' that intersect the given line in the hyperbolic plane, and '''ultra parallel geodesics''' that do not intersect even in the limit to infinity (in either direction). In the Euclidean limit the latter set is empty. |
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