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Line segment
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a line segment is a part of a line
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
 that is bounded by two end points
Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue....
, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
, the line segment is either an edge
Edge (geometry)

In geometry, an edge is a one-dimensional line segment joining two zero-dimensional vertex in a polytope. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....
 (of that polygon) if they are adjacent vertices, or otherwise a diagonal
Diagonal

A diagonal can refer to a line joining two nonconsecutive vertices of a polygon or polyhedron, or in informal contexts any upward or downward sloping line....
. When the end points both lie on a curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
 such as a circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
, a line segment is called a chord
Chord (geometry)

A chord of a curve is a geometry line segment whose endpoints both lie on the curve.A secant or a secant line is the line extension of a chord....
 (of that curve).
is a vector space
Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
 over or , and is a subset
Subset

In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
 of then is a line segment if can be parameterized as

for some vectors , in which case the vectors and are called the end points of

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






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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a line segment is a part of a line
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
 that is bounded by two end points
Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue....
, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
, the line segment is either an edge
Edge (geometry)

In geometry, an edge is a one-dimensional line segment joining two zero-dimensional vertex in a polytope. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....
 (of that polygon) if they are adjacent vertices, or otherwise a diagonal
Diagonal

A diagonal can refer to a line joining two nonconsecutive vertices of a polygon or polyhedron, or in informal contexts any upward or downward sloping line....
. When the end points both lie on a curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
 such as a circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
, a line segment is called a chord
Chord (geometry)

A chord of a curve is a geometry line segment whose endpoints both lie on the curve.A secant or a secant line is the line extension of a chord....
 (of that curve).

Definition

If is a vector space
Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
 over or , and is a subset
Subset

In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
 of then is a line segment if can be parameterized as

for some vectors , in which case the vectors and are called the end points of

Sometimes one needs to distinguish between "open" and "closed" line segments. Then one defines a closed line segment as above, and an open line segment as a subset that can be parametrized as

for some vectors .

An alternative, equivalent, definition is as follows: A (closed) line segment is a convex hull
Convex hull

In mathematics, the convex hull or convex envelope for a Set of points X in a real vector space V is the minimal convex set containing X....
 of two points.

Properties

  • A line segment is a connected, non-empty set.
  • If is a topological vector space
    Topological vector space

    In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. As the name suggests the space blends a topology with the algebraic concept of a vector space....
    , then a closed line segment is a closed set
    Closed set

    In topology and related branches of mathematics, a closed set is a Set whose complement is open set....
     in However, an open line segment is an open set in if and only if
    If and only if

    If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a biconditional logical connective between statements....
      is one-dimensional.
  • More generally than above, the concept of a line segment can be defined in an ordered geometry
    Ordered geometry

    Ordered geometry is a form of geometry featuring the concept of intermediacy but, like projective geometry, omitting the basic notion of measurement....
    .


In Proofs


In geometry, the segment addition postulate states that if B is between A and C, then segment AB + segment BC = segment AC.

See also

  • Interval (mathematics)
    Interval (mathematics)

    In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set....
  • Line (geometry)


External links

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