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Concept
Line (geometry)
Summary
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist embedded in two, three, or higher dimensional spaces. The word line may also refer to a line segment in everyday life that has two points to denote its ends (endpoints). A line can be referred to by two points that lie on it (e.g. ) or by a single letter (e.g. ).
Euclid described a line as a "breadthless length" that "lies evenly with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed all of geometry. Euclidean line and Euclidean geometry are terms introduced to avoid confusion with generalizations introduced since the end of the 19th century, such as non-Euclidean, projective, and affine geometry.
When geometry was first formalised by Euclid in the Elements, he defined a general line (now called a curve) to be "breadthless length" with a straight line being a line "which lies evenly with the points on itself". These definitions serve little purpose since they use terms that are not themselves defined. In fact, Euclid himself did not use these definitions in this work and probably included them just to make it clear to the reader what was being discussed. In modern geometry, a line is simply taken as an undefined object with properties given by axioms, but it is sometimes defined as a set of points obeying a linear relationship when some other fundamental concept is left undefined.
In an axiomatic formulation of Euclidean geometry, such as that of Hilbert (Euclid's original axioms contained various flaws that have been corrected by modern mathematicians), a line is stated to have certain properties that relate it to other lines and points. For example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect at most at one point. In two dimensions (i.e., the Euclidean plane), two lines that do not intersect are called parallel.
Official source
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