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# Line

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The word line apparently derives from the Latin linum, meaning flax plant from which linen is produced; at one time, a stretched linen thread was the most reliable way to determine a straight line. Also see liner[?] and lining[?].

In telecommunications, a telephone line is a single-user circuit on a telephone system. More generally, a line is a circuit or loop in any communications system.

A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object. Given two points, one can always find exactly one line that passes through the two points; the line provides the shortest connection between the points. Two different lines can intersect in at most one point; two different planes can intersect in at most one line. This intuitive concept of a line can be formalized in various ways.

If geometry is developed axiomatically (as in Euclid's Elements and later in David Hilbert's Foundations of Geometry[?]), then lines are not defined at all, but characterized axiomatically by their properties. "Everything that satisfies the axioms for a line is a line." While Euclid did define a line as "length without breadth", he did not use this rather obscure definition in his later development.

In Euclidean space Rn (and analogously in all other vector spaces), we define a line L as a subset of the form

$L = \{\mathbf{a}+t\mathbf{b}\mid t\in\mathbb{R}\}$

where a and b are given vectors in Rn with b non-zero. The vector b describes the direction of the line, and a is a point on the line. Different choices of a and b can yield the same line.

One can show that in R2, every line L is described by a linear equation of the form

$L=\{(x,y)\mid ax+by=c\}$

with fixed real coefficients a, b and c such that a and b are not both zero. An important property of these lines is their slope.

More abstractly, one usually thinks of the real line as the prototype of a line, and assumes that the points on a line stand in a one-to-one correspondence with the real numbers. However, one could also use the hyperreal numbers for this purpose, or even the long line of topology.

The "straightness" of a line, interpreted as the property that it minimizes distances between its points, can be generalized and leads to the concept of geodesics on differentiable manifolds.

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