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Since these are equivalent properties, any one of them could be taken as the definition of parallel lines in Euclidean space, but the first and third properties involve measurement, and so, are "more complicated" than the second. Thus, the second property is the one usually chosen as the defining property of parallel lines in Euclidean geometry ...
Euclid gave the definition of parallel lines in Book I, Definition 23 [2] just before the five postulates. [3] Euclidean geometry is the study of geometry that satisfies all of Euclid's axioms, including the parallel postulate. The postulate was long considered to be obvious or inevitable, but proofs were elusive.
In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.
Others [3] define a trapezoid as a quadrilateral with at least one pair of parallel sides (the inclusive definition [7]), making the parallelogram a special type of trapezoid. The latter definition is consistent with its uses in higher mathematics such as calculus. This article uses the inclusive definition and considers parallelograms as ...
Hence by using curves in parallel coordinates instead of lines, the point line duality is lost together with all the other properties of projective geometry, and the known nice higher-dimensional patterns corresponding to (hyper)planes, curves, several smooth (hyper)surfaces, proximities, convexity and recently non-orientability. [6]
However, parallel (non-crossing) pairs of lines are less restricted in hyperbolic line arrangements than in the Euclidean plane: in particular, the relation of being parallel is an equivalence relation for Euclidean lines but not for hyperbolic lines. [51] The intersection graph of the lines in a hyperbolic arrangement can be an arbitrary ...
In elliptic geometry, two lines are Clifford parallel or paratactic lines if the perpendicular distance between them is constant from point to point. The concept was first studied by William Kingdon Clifford in elliptic space and appears only in spaces of at least three dimensions.
Parallel curves of the graph of = for distances =, …, Two definitions of a parallel curve: 1) envelope of a family of congruent circles, 2) by a fixed normal distance The parallel curves of a circle (red) are circles, too