Line segment


In geometry, a line segment is a part of a straight line that is bounded by two distinct endpoints, and contains every point on the line that is between its endpoints. It is a special case of an arc, with zero curvature. The length of a line segment is given by the Euclidean distance 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. In geometry, a line segment is often denoted using an overline above the symbols for the two endpoints, such as in.
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 a diagonal. When the end points both lie on a curve, a line segment is called a chord.

In real or complex vector spaces

If is a vector space over or and is a subset of, then is a line segment if can be parameterized as
for some vectors where is nonzero. The endpoints of are then the vectors and.
Sometimes, one needs to distinguish between "open" and "closed" line segments. In this case, one would define a closed line segment as above, and an open line segment as a subset 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, one might define point to be between two other points and, if the distance added to the distance is equal to the distance. Thus in the line segment with endpoints and is the following collection of points:

Properties

In an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or defined in terms of an isometry of a line.
Segments play an important role in other theories. For example, in a convex set, 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.

As a degenerate ellipse

A line segment can be viewed as a degenerate case of an ellipse, in which the semiminor axis goes to zero, the foci go 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 other geometric shapes

In addition to appearing as the edges and diagonals of polygons and polyhedra, line segments also appear in numerous other locations relative to other geometric shapes.

Triangles

Some very frequently considered segments in a triangle to 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 is called a directed line segment or oriented line segment. It suggests a translation or displacement. The magnitude and direction are indicative of a potential change. Extending a directed line segment semi-infinitely produces a directed half-line and infinitely in both directions produces a directed line. 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 equipollent any pair having the same length and orientation. This application of an equivalence relation was introduced by Giusto Bellavitis in 1835.

Generalizations

Analogous to straight line segments above, one can also define arcs as segments of a curve.
In one-dimensional space, a ball is a line segment.
An oriented plane segment or bivector generalizes the directed line segment.
Beyond Euclidean geometry, geodesic segments play the role of line segments.
A line segment is a one-dimensional simplex; a two-dimensional simplex is a triangle.

Types of line segments