A **line**, or **straight line**, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i.e. a curve that is long and straight. Given two points, in Euclidean geometry, one can always find exactly one line that passes through the two points; the line provides the shortest connection between the points. Three or more points that lie on the same line are called **collinear**. 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 **R**^{n} (and analogously in all other vector spaces), we define a line *L* as a subset of the form where **a** and **b** are given vectors in **R**^{n} 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 **R**^{2}, every line *L* is described by a linear equation of the form with fixed real coefficients *a*, *b* and *c* such that *a* and *b* are not both zero. Important properties of these lines are their slope, x-intercept and y-intercept. 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. |