Part of a line that is bounded by two distinct end points; line with two endpoints
The geometric definition of a closed line segment: theintersection of all points at or to the right ofA with all points at or to the left ofBHistorical image of 1699 - creating a line segment
Ingeometry, aline segment is a part of astraight line that is bounded by two distinctendpoints (itsextreme points), and contains everypoint on the line that is between its endpoints. It is a special case of anarc, with zerocurvature. Thelength of a line segment is given by theEuclidean distance between its endpoints. Aclosed line segment includes both endpoints, while anopen line segment excludes both endpoints; ahalf-open line segment includes exactly one of the endpoints. Ingeometry, a line segment is often denoted using anoverline (vinculum) above the symbols for the two endpoints, such as inAB.[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 apolygon orpolyhedron, the line segment is either anedge (of that polygon or polyhedron) if they are adjacent vertices, or adiagonal. When the end points both lie on acurve (such as acircle), a line segment is called achord (of that curve).
IfV is avector space over or andL is asubset ofV, thenL is aline segment ifL can be parameterized as
for some vectors wherev is nonzero. The endpoints ofL are then the vectorsu andu +v.
Sometimes, one needs to distinguish between "open" and "closed" line segments. In this case, one would define aclosed line segment as above, and anopen line segment as a subsetL that can be parametrized as
for some vectors
Equivalently, a line segment is theconvex hull of two points. Thus, the line segment can be expressed as aconvex combination of the segment's two end points.
Ingeometry, one might define pointB to be between two other pointsA andC, 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:
More generally than above, the concept of a line segment can be defined in anordered 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 anisometry of a line (used as a coordinate system).
Segments play an important role in other theories. For example, in aconvex 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. Thesegment 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 adegenerate case of anellipse, in which the semiminor axis goes to zero, thefoci 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 aradial elliptic trajectory.
Some very frequently considered segments in atriangle to include the threealtitudes (eachperpendicularly connecting a side or itsextension to the oppositevertex), the threemedians (each connecting a side'smidpoint to the opposite vertex), theperpendicular bisectors of the sides (perpendicularly connecting the midpoint of a side to one of the other sides), and theinternal angle bisectors (each connecting a vertex to the opposite side). In each case, there are variousequalities relating these segment lengths to others (discussed in the articles on the various types of segment), as well asvarious inequalities.
In addition to the sides and diagonals of aquadrilateral, some important segments are the twobimedians (connecting the midpoints of opposite sides) and the fourmaltitudes (each perpendicularly connecting one side to the midpoint of the opposite side).
Any straight line segment connecting two points on acircle orellipse is called achord. Any chord in a circle which has no longer chord is called adiameter, and any segment connecting the circle'scenter (the midpoint of a diameter) to a point on the circle is called aradius.
In an ellipse, the longest chord, which is also the longestdiameter, is called themajor axis, and a segment from the midpoint of the major axis (the ellipse's center) to either endpoint of the major axis is called asemi-major axis. Similarly, the shortest diameter of an ellipse is called theminor axis, and the segment from its midpoint (the ellipse's center) to either of its endpoints is called asemi-minor axis. The chords of an ellipse which areperpendicular to the major axis and pass through one of itsfoci are called thelatera recta of the ellipse. Theinterfocal segment connects the two foci.
When a line segment is given anorientation (direction) it is called adirected line segment ororiented line segment. It suggests atranslation ordisplacement (perhaps caused by aforce). The magnitude and direction are indicative of a potential change. Extending a directed line segment semi-infinitely produces adirected half-line and infinitely in both directions produces adirected line. This suggestion has been absorbed intomathematical physics through the concept of aEuclidean vector.[2][3] The collection of all directed line segments is usually reduced by makingequipollent any pair having the same length and orientation.[4] This application of anequivalence relation was introduced byGiusto Bellavitis in 1835.
