Euclid defined the line, then never used his definition again
Euclid's Elements calls a straight line a breadthless length that lies evenly with its own points. Those words lean on the reader's physical intuition and undefined terms, and the rest of the book never cites them. Modern geometry simply treats the line as a basic notion governed by axioms, or as points satisfying a linear equation.
A line in geometry has infinite extent and no width, depth or bend. It idealises a straightedge, a taut string or a light ray, and it is one-dimensional, able to sit inside planes or spaces of three or more dimensions. Everyday speech often uses the word for a segment, the piece between two endpoints. In Euclid's own terms, a general line was what we now call a curve and a straight line was closer to what we now call a segment. The labels Euclidean line and Euclidean geometry appeared later, to separate the classical picture from non-Euclidean, projective and affine geometries developed from the late 19th century.
David Hilbert and others added axioms to Euclid's to close logical gaps. In such a system, two distinct points always determine exactly one line, and two distinct lines meet at most once. In the plane, lines that never meet are parallel. In higher dimensions non-meeting lines are parallel only if some plane contains both; otherwise they are skew.
Algebra gives several handles on a line. In three dimensions one first-degree equation in x, y and z describes a plane, so two such equations, for planes that are not parallel, pick out their line of intersection; in n dimensions it generally takes n minus 1 equations. Given two points a and b, the line through them is every combination (1 minus t) times a plus t times b for real t, and reading it from a toward b gives it a direction. A directed line with a special role, such as the one a body spins about, is called an axis.
Points that share a line are collinear, and three points that are not collinear fix exactly one plane. In coordinates, three points are collinear when a matrix built from them has rank below three; in the plane that means its determinant is zero. A line also splits the plane into two regions, and any finite set of lines carves the plane into convex pieces, some unbounded, called an arrangement of lines.
Source: Line (geometry)
