Line segment

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intersection
of all points at or to the right of A with all points at or to the left of B
historical image – create a line segment (1699)

In

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 (vinculum) above the symbols for the two endpoints, such as in AB.[1]

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

In real or complex vector spaces

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 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

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 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 the line segment with endpoints and is the following collection of points:

Properties

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 isometry of a line (used as a coordinate system).

Segments play an important role in other theories. For example, in a

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

geometric shapes
.

Triangles

Some very frequently considered segments in a

internal angle bisectors (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

orthocenter
.

Quadrilaterals

In addition to the sides and diagonals of a quadrilateral, 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

center (the midpoint of a diameter) to a point on the circle is called a radius
.

In an ellipse, the longest chord, which is also the longest

latera recta
of the ellipse. The interfocal segment connects the two foci.

Directed line segment

When a line segment is given an

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.

Generalizations

Analogous to

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.

Types of line segments

See also

Notes

  1. ^ "Line Segment Definition - Math Open Reference". www.mathopenref.com. Retrieved 2020-09-01.

References

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

External links

This article incorporates material from Line segment on

Creative Commons Attribution/Share-Alike License
.