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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 ...
Parallel (geometry), two lines in the Euclidean plane which never intersect Parallel (operator) , mathematical operation named after the composition of electrical resistance in parallel circuits Science and engineering
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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.
Parallel computing, the simultaneous execution on multiple processors of different parts of a program In the analysis of parallel algorithms, the maximum possible speedup of a computation; Parallel evolution, the independent emergence of a similar trait in different unrelated species; Parallel (geometry), the property of parallel lines
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]
In this case, one gets a parallel curve on the opposite side of the curve (see diagram on the parallel curves of a circle). One can easily check that a parallel curve of a line is a parallel line in the common sense, and the parallel curve of a circle is a concentric circle.
Another type of non-Euclidean geometry is the hyperbolic plane, and arrangements of lines in this geometry have also been studied. [50] Any finite set of lines in the Euclidean plane has a combinatorially equivalent arrangement in the hyperbolic plane (e.g. by enclosing the vertices of the arrangement by a large circle and interpreting the ...