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# Line segment ( Elementary Geometry )

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Rank: 100%     In , a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The 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 , a line segment is often denoted using a line above the symbols for the two endpoints (such as ).

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 or , the line segment is either an edge (of that polygon or polyhedron) if they are adjacent vertices, or a . When the end points both lie on a (such as a ), a line segment is called a chord (of that curve).

In real or complex vector spaces
If is a over or and is a of , then is a line segment if can be parameterized as
$L = \\left\{ \mathbf\left\{u\right\} + t\mathbf\left\{v\right\} \mid t \in 0,1\\right\}$

for some vectors $\mathbf\left\{u\right\}, \mathbf\left\{v\right\} \in V.$ In which case, the vectors and are called the end points of .

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

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

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

Equivalently, a line segment is the of two points. Thus, the line segment can be expressed as a convex combination of the segment's two end points.

In , 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 $A=\left(a_x,a_y\right)$ and $C=\left(c_x,c_y\right)$ is the following collection of points:

$\Biggl\\left\{ \left(x,y\right) \mid \sqrt\left\{\left(x-c_x\right)^2 + \left(y-c_y\right)^2\right\} + \sqrt\left\{\left(x-a_x\right)^2 + \left(y-a_y\right)^2\right\} = \sqrt\left\{\left(c_x-a_x\right)^2 + \left(c_y-a_y\right)^2\right\} \Biggr\\right\} .$

Properties
• A line segment is a , set.
• If is a topological vector space, then a closed line segment is a in . However, an open line segment is an in if and only if is one-dimensional.
• More generally than above, the concept of a line segment can be defined in an .
• A pair of line segments can be any one of the following: intersecting, parallel, , 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 proofs
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 of a line (used as a coordinate system).

Segments play an important role in other theories. For example, in a , 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 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 of and , line segments also appear in numerous other locations relative to other .

Triangles
Some very frequently considered segments in a to include the three altitudes (each connecting a side or its to the opposite vertex), the three medians (each connecting a side's to the opposite vertex), the perpendicular bisectors of the sides (perpendicularly connecting the midpoint of a side to one of the other sides), and the (each connecting a vertex to the opposite side). In each case, there are various equalities relating these segment lengths to others (discussed in the articles on the various types of segment), as well as various inequalities.

Other segments of interest in a triangle include those connecting various to each other, most notably the , the , the nine-point center, the and the .

In addition to the sides and diagonals of a , some important segments are the two bimedians (connecting the midpoints of opposite sides) and the four maltitudes (each perpendicularly connecting one side to the midpoint of the opposite side).

Circles and ellipses
Any straight line segment connecting two points on a or is called a chord. Any chord in a circle which has no longer chord is called a , and any segment connecting the circle's center (the midpoint of a diameter) to a point on the circle is called a .

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 (the ellipse's center) 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 (the ellipse's center) to either of its endpoints is called a semi-minor axis. The chords of an ellipse which are to the major axis and pass through one of its foci are called the of the ellipse. The interfocal segment connects the two foci.

Directed line segment
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 ). 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 .Harry F. Davis & Arthur David Snider (1988) Introduction to Vector Analysis, 5th edition, page 1, Wm. C. Brown Publishers Matiur Rahman & Isaac Mulolani (2001) Applied Vector Analysis, pages 9 & 10, The collection of all directed line segments is usually reduced by making "equivalent" any pair having the same length and orientation.Eutiquio C. Young (1978) Vector and Tensor Analysis, pages 2 & 3, 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 segments above, one can also define arcs as segments of a .

In one-dimensional space, a ball is a line segment.

Types of line segments

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

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

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