Line segment

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Not to be confused with arc (geometry). Line segment_sentence_0

Examples of line segments include the sides of a triangle or square. Line segment_sentence_1

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. Line segment_sentence_2

When the end points both lie on a curve (such as a circle), a line segment is called a chord (of that curve). Line segment_sentence_3

In real or complex vector spaces Line segment_section_0

Sometimes, one needs to distinguish between "open" and "closed" line segments. Line segment_sentence_4

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 Line segment_sentence_5

Equivalently, a line segment is the convex hull of two points. Line segment_sentence_6

Thus, the line segment can be expressed as a convex combination of the segment's two end points. Line segment_sentence_7

Properties Line segment_section_1

Line segment_unordered_list_0

  • A line segment is a connected, non-empty set.Line segment_item_0_0
  • 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.Line segment_item_0_1
  • More generally than above, the concept of a line segment can be defined in an ordered geometry.Line segment_item_0_2
  • 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.Line segment_item_0_3

In proofs Line segment_section_2

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). Line segment_sentence_8

Segments play an important role in other theories. Line segment_sentence_9

For example, a set is convex if the segment that joins any two points of the set is contained in the set. Line segment_sentence_10

This is important because it transforms some of the analysis of convex sets, to the analysis of a line segment. Line segment_sentence_11

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. Line segment_sentence_12

As a degenerate ellipse Line segment_section_3

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. Line segment_sentence_13

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. Line segment_sentence_14

A complete orbit of this ellipse traverses the line segment twice. Line segment_sentence_15

As a degenerate orbit, this is a radial elliptic trajectory. Line segment_sentence_16

In other geometric shapes Line segment_section_4

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. Line segment_sentence_17

Triangles Line segment_section_5

Some very frequently considered segments in a triangle to include the three altitudes (each perpendicularly connecting a side or its extension to the opposite vertex), the three medians (each connecting a side's midpoint 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 internal angle bisectors (each connecting a vertex to the opposite side). Line segment_sentence_18

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. Line segment_sentence_19

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. Line segment_sentence_20

Quadrilaterals Line segment_section_6

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). Line segment_sentence_21

Circles and ellipses Line segment_section_7

Any straight line segment connecting two points on a circle or ellipse is called a chord. Line segment_sentence_22

Any chord in a circle which has no longer chord is called a diameter, and any segment connecting the circle's center (the midpoint of a diameter) to a point on the circle is called a radius. Line segment_sentence_23

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. Line segment_sentence_24

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. Line segment_sentence_25

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. Line segment_sentence_26

The interfocal segment connects the two foci. Line segment_sentence_27

Directed line segment Line segment_section_8

When a line segment is given an orientation (direction) it suggests a translation or perhaps a force tending to make a translation. Line segment_sentence_28

The magnitude and direction are indicative of a potential change. Line segment_sentence_29

This suggestion has been absorbed into mathematical physics through the concept of a Euclidean vector. Line segment_sentence_30

The collection of all directed line segments is usually reduced by making "equivalent" any pair having the same length and orientation. Line segment_sentence_31

This application of an equivalence relation dates from Giusto Bellavitis’s introduction of the concept of equipollence of directed line segments in 1835. Line segment_sentence_32

Generalizations Line segment_section_9

Analogous to straight line segments above, one can also define arcs as segments of a curve. Line segment_sentence_33

See also Line segment_section_10

Line segment_unordered_list_1

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: segment.