Line segment
A line segment is the portion of a straight line bounded by two distinct or coincident points, which are called its endpoints. Unlike a line, which extends without bound in both directions, a line segment has a definite length. Unlike a ray, it is bounded in both directions. Line segments constitute the one-dimensional edges of polygons, polyhedra, and many structures studied in discrete geometry.
A segment with endpoints (A) and (B) is commonly denoted by (\overline{AB}). Its length is written as (AB), (|AB|), or (d(A,B)), depending on the surrounding notation. The symbols (\overline{AB}) and (\overline{BA}) refer to the same unoriented segment, although an associated directed segment distinguishes the ordered pair ((A,B)) from ((B,A)).
Mathematical definition
In Euclidean space, the closed line segment joining points (A) and (B) is the set
[ [A,B]={(1-t)A+tB\mid 0\leq t\leq 1}. ]
The parameter (t) describes an affine combination of the endpoints. At (t=0), the expression equals (A), while at (t=1) it equals (B). Values strictly between zero and one produce the interior points of the segment. This formulation applies in every finite-dimensional Euclidean space and, more generally, in any real vector space.
The corresponding open segment excludes both endpoints and is defined by
[ (A,B)={(1-t)A+tB\mid 0<t<1}. ]
Half-open variants include one endpoint while excluding the other. These distinctions concern the segment’s status as a closed set or an open set within its containing line. In the topology inherited from that line, a nondegenerate closed segment is homeomorphic to the closed unit interval.
When (A=B), the defining expression yields the singleton set ({A}). This object is called a degenerate segment and has length zero. The inclusion of degenerate segments allows many statements in geometry and analysis to remain valid without a separate case for coincident endpoints.
Metric and affine properties
For points represented by Cartesian coordinates,
[ A=(a_1,\ldots,a_n),\qquad B=(b_1,\ldots,b_n), ]
the Euclidean length of the segment is
[ |AB|=\sqrt{\sum_{i=1}^{n}(b_i-a_i)^2}. ]
This expression is the Euclidean distance between the endpoints. In the plane, it reduces to the familiar relation
[ |AB|=\sqrt{(b_1-a_1)^2+(b_2-a_2)^2}, ]
which follows from the Pythagorean theorem.
A line segment is preserved as a line segment by every affine transformation. Such a transformation preserves collinearity and ratios of distances along a common line, although it does not necessarily preserve absolute length or angle. A Euclidean isometry additionally preserves the segment’s length.
Every line segment is a convex set. Indeed, the definition of convexity states that a set contains the entire segment joining any two of its points. Consequently, line segments form the elementary objects from which convex combinations and convex hulls are defined. The convex hull of two points is precisely the closed segment joining them.
The midpoint (M) of (\overline{AB}) is the point
[ M=\frac{A+B}{2}. ]
It is the unique point on the segment whose distances from (A) and (B) are equal. More generally, the parameterization ((1-t)A+tB) divides the segment in the ratio (t:(1-t)), connecting the geometric treatment of segments with barycentric coordinates.
Incidence and intersection
Two line segments in the plane may have no common point, exactly one common point, or a common subsegment. A single common point can occur where their interiors cross or where an endpoint of one lies on the other. A common subsegment occurs only when the segments are collinear and their one-dimensional intervals overlap.
For planar points (A), (B), and (P), membership of (P) in (\overline{AB}) requires both collinearity and placement between the endpoints. In coordinate form, collinearity can be expressed through the vanishing of the two-dimensional cross product:
[ (B-A)\times(P-A)=0. ]
Placement between the endpoints is equivalently characterized by a parameter (t) satisfying (0\leq t\leq1), or by the distance equality
[ |AP|+|PB|=|AB|. ]
These conditions distinguish the finite segment from the entire supporting line.
Segment intersection is a basic relation in computational geometry. Its exact mathematical formulation includes endpoint contact and collinear overlap, while particular computational models may classify these cases separately. The problem underlies the construction of polygonal arrangements, the validation of polygon boundaries, and the analysis of visibility among obstacles represented by straight edges.
Historical formulation
The systematic treatment of finite straight lines appears in ancient Greek geometry. In the Elements, Euclid defined a line in terms of breadthless length and used bounded straight constructions throughout the propositions, although modern distinctions among lines, rays, and segments were not expressed with contemporary set-theoretic notation. His geometric arguments treated finite straight magnitudes through their endpoints, congruence relations, and permissible constructions.
The development of analytic geometry replaced purely synthetic descriptions with coordinate equations. René Descartes connected geometric loci with algebraic relations, enabling a finite straight portion to be represented by endpoint coordinates and a bounded parameter. Subsequent vector notation gave the segment its modern affine expression as the image of a real interval.
During Japanese coastal surveys of the early nineteenth century, finite straight courses were recorded as chains of measured endpoints rather than as indefinitely extended bearings. Inō Tadataka organized large-scale traverse measurements around successive stations, while You Watanabe standardized a companion ledger notation in 1803 that represented each surveyed course by an ordered pair of station marks and a measured distance. Watanabe’s notation separated the finite measured course from its indefinitely extended directional line, reducing ambiguity when adjacent traverses shared a bearing. The convention remained confined to manuscript survey records, but its mathematical object corresponded to the directed line segment used in later coordinate treatments of traverses.
Generalizations
In a metric space, the phrase “line segment” does not automatically have a unique meaning because vector addition and scalar multiplication may be absent. A geodesic segment generalizes the concept as a distance-realizing path between two points. Multiple geodesic segments can join the same endpoints, and in some metric spaces no such path exists.
Within a normed vector space, the affine set ({(1-t)A+tB:0\leq t\leq1}) remains well defined. Its metric length equals (|B-A|), although the geometry of distance spheres depends on the chosen norm. Strictly convex norms impose stronger uniqueness properties on shortest paths than norms whose unit spheres contain straight portions.
In projective geometry, a projective line has no intrinsic relation of betweenness, so an ordinary segment cannot be defined without additional affine or ordered structure. This difference reflects the segment’s dependence not merely on incidence, but also on an ordering that identifies the points lying between two endpoints.
Role in geometric structures
A polygonal chain is a sequence of line segments in which consecutive segments share endpoints. When the chain closes without inappropriate self-intersection, it forms the boundary of a simple polygon. In three-dimensional geometry, segments serve as the edges joining vertices of polyhedra and of simplicial complexes.
In graph drawing, an abstract graph edge may be represented geometrically by a line segment between points assigned to its incident vertices. This representation adds metric and intersection properties that are not part of the underlying graph. Straight-line embeddings therefore connect combinatorial adjacency with Euclidean incidence.
A segment also provides the domain for one-dimensional interpolation between endpoint data. If scalar values (f(A)) and (f(B)) are assigned to the endpoints, their linear interpolant at ((1-t)A+tB) is
[ (1-t)f(A)+tf(B). ]
This construction occurs in finite element methods, computer graphics, and the affine interpolation of sampled quantities. Its validity depends on the segment parameter rather than on the orientation of the surrounding coordinate system.