Line segment

If V is a vector space over ${\displaystyle \mathbb {R} }$ or ${\displaystyle \mathbb {C} ,}$ and L is a subset of V, then L is a line segment if L can be parameterized as

${\displaystyle L=\{\mathbf {u} +t\mathbf {v} \mid t\in [0,1]\}}$

for some vectors ${\displaystyle \mathbf {u} ,\mathbf {v} \in V}$ where v is nonzero. The endpoints of L are then the vectors u and u + v.

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 L that can be parametrized as

${\displaystyle L=\{\mathbf {u} +t\mathbf {v} \mid t\in (0,1)\}}$

for some vectors ${\displaystyle \mathbf {u} ,\mathbf {v} \in V.}$

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 B to be between two other points A and C, if the distance |AB| added to the distance |BC| is equal to the distance |AC|. Thus in ${\displaystyle \mathbb {R} ^{2},}$ the line segment with endpoints ${\displaystyle A=(a_{x},a_{y})}$ and ${\displaystyle C=(c_{x},c_{y})}$ is the following collection of points:

${\displaystyle {\Biggl \{}(x,y)\mid {\sqrt {(x-c_{x})^{2}+(y-c_{y})^{2}}}+{\sqrt {(x-a_{x})^{2}+(y-a_{y})^{2}}}={\sqrt {(c_{x}-a_{x})^{2}+(c_{y}-a_{y})^{2}}}{\Biggr \}}.}$

• A line segment is a connected, non-empty set.
• 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 then they must cross each other, but that need not be true of segments.

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 (used as a coordinate system).

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.

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 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.

When a line segment is given an orientation (direction) it is called a directed line segment. It suggests a translation or displacement (perhaps caused by a force). The magnitude and direction are indicative of a potential change. Extending a directed line segment semi-infinitely produces a ray and infinitely in both directions produces a directed line. This suggestion has been absorbed into mathematical physics through the concept of a Euclidean vector.[2][3] The collection of all directed line segments is usually reduced by making "equivalent" any pair having the same length and orientation.[4] This application of an equivalence relation dates from Giusto Bellavitis's introduction of the concept of equipollence of directed line segments in 1835.

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.

• Chord (geometry)
• Diameter
• Radius

• Polygonal chain
• Interval (mathematics)
• Line segment intersection, the algorithmic problem of finding intersecting pairs in a collection of line segments

• David Hilbert The Foundations of Geometry. The Open Court Publishing Company 1950, p. 4

Wikimedia Commons has media related to Line segment.

Look up line segment in Wiktionary, the free dictionary.

• Weisstein, Eric W. "Line segment". MathWorld.
• Line Segment at PlanetMath
• Copying a line segment with compass and straightedge
• Dividing a line segment into N equal parts with compass and straightedge Animated demonstration

This article incorporates material from Line segment on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.