In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge if they are adjacent vertices, or otherwise a diagonal. When the end points both lie on a curve such as a circle, a line segment is called a chord.
If V is a vector space over or, and L is a subset of V, then L is a line segment if L can be parameterized as for some vectors, in which case the vectors u and are called the end points of L. Sometimes one needs to distinguish between "open" and "closed" line segments. Then one defines a closed line segment as above, and an open line segment as a subset L that can be parametrized as for some vectors. Equivalently, a line segment is the convex hull of two points. Thus, the line segment can be expressed as a convex combination of the segment's two end points. In geometry, it is sometimes defined that a point B is between two other points A and C, if the distanceAB added to the distance BC is equal to the distance AC. Thus in the line segment with endpoints and is the following collection of points:
If V is a topological vector space, then a closed line segment is a closed set in V. However, an open line segment is an open set in V if and only if V is one-dimensional.
More generally than above, the concept of a line segment can be defined in an ordered geometry.
A pair of line segments can be any one of the following: intersecting, parallel, skew, or none of these. The last possibility is a way that line segments differ from lines: if two nonparallel lines are in the same Euclidean plane they must cross each other, but that need not be true of segments.
In proofs
In an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or else be defined in terms of an isometry of a line. Segments play an important role in other theories. For example, a set is convex if the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of convex sets to the analysis of a line segment. The Segment Addition Postulate can be used to add congruent segment or segments with equal lengths and consequently substitute other segments into another statement to make segments congruent.
A line segment can be viewed as a degenerate case of an ellipse in which the semiminor axis goes to zero, the focigo to the endpoints, and the eccentricity goes to one. A standard definition of an ellipse is the set of points for which the sum of a point's distances to two foci is a constant; if this constant equals the distance between the foci, the line segment is the result. A complete orbit of this ellipse traverses the line segment twice. As a degenerate orbit this is a radial elliptic trajectory.
In addition to appearing as the edges and diagonals of polygons and polyhedra, line segments appear in numerous other locations relative to other geometric shapes.
Triangles
Some very frequently considered segments in a triangle include the three altitudes, the three medians, the perpendicular bisectors of the sides, and the internal angle bisectors. In each case there are various equalities relating these segment lengths to others as well as various inequalities. Other segments of interest in a triangle include those connecting various triangle centers to each other, most notably the incenter, the circumcenter, the nine-point center, the centroid, and the orthocenter.
Quadrilaterals
In addition to the sides and diagonals of a quadrilateral, some important segments are the two bimedians and the four maltitudes.
Circles and ellipses
Any straight line segment connecting two points on a circle or ellipse is called a chord. Any chord in a circle which has no longer chord is called a diameter, and any segment connecting the circle's center to a point on the circle is called a radius. In an ellipse, the longest chord, which is also the longest diameter, is called the major axis, and a segment from the midpoint of the major axis to either endpoint of the major axis is called a semi-major axis. Similarly, the shortest diameter of an ellipse is called the minor axis, and the segment from its midpoint to either of its endpoints is called a semi-minor axis. The chords of an ellipse which are perpendicular to the major axis and pass through one of its foci are called the latera recta of the ellipse. The interfocal segment connects the two foci.
Directed line segment
When a line segment is given an orientation it suggests a translation or perhaps a force tending to make a translation. The magnitude and direction are indicative of a potential change. This suggestion has been absorbed into mathematical physics through the concept of a Euclidean vector. The collection of all directed line segments is usually reduced by making "equivalent" any pair having the same length and orientation. This application of an equivalence relation dates from Giusto Bellavitis’s introduction of the concept of equipollence of directed line segments in 1835.
Generalizations
Analogous to straight line segments above, one can define arcs as segments of a curve.
