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Line art drawing of parallel lines and curves. In geometry, parallel lines are coplanar infinite straight lines that do not intersect at any point. Parallel planes are planes in the same three-dimensional space that never meet. Parallel curves are curves that do not touch each other or intersect and keep a fixed minimum distance. In three ...
This postulate does not specifically talk about parallel lines; [1] it is only a postulate related to parallelism. 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.
In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted or . It is a geometric space in which two real numbers are required to determine the position of each point . It is an affine space , which includes in particular the concept of parallel lines .
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 interior of the circle as a Klein model of the hyperbolic plane). However, parallel (non-crossing) pairs of lines are less restricted in ...
The attitude of a lattice plane is the orientation of the line normal to the plane, [12] and is described by the plane's Miller indices. In three-space a family of planes (a series of parallel planes) can be denoted by its Miller indices (hkl), [13] [14] so the family of planes has an attitude common to all its constituent planes.
In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point. [ 1 ] It is equivalent to Euclid's parallel postulate in the context of Euclidean geometry [ 2 ] and was named after the Scottish mathematician John Playfair .
Parallel lines in the Euclidean plane are said to intersect at a point at infinity corresponding to their common direction. Given a point (,) on the Euclidean plane, for any non-zero real number , the triple (,,) is called a set of homogeneous coordinates for the point. By this definition, multiplying the three homogeneous coordinates by a ...
The Euclidean plane follows Euclidean geometry, and in particular the parallel postulate. A projective plane may be constructed by adding "points at infinity" where two otherwise parallel lines would intersect, so that every pair of lines intersects in exactly one point.