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Encyclopedia > Line segment
The geometric definition of a line segment
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The geometric definition of a line segment

In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Image File history File links Line_segment_-_definition. ... Image File history File links Line_segment_-_definition. ... For other uses, see Geometry (disambiguation). ... A line, or straight line, can be described as an (infinitely) thin, (infinitely) long, perfectly straight curve (the term curve in mathematics includes straight curves). In Euclidean geometry, exactly one line can be found that passes through any two points. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ...


Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment is either an edge (of that polygon) if they are adjacent vertices, or otherwise 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). Look up Polygon in Wiktionary, the free dictionary. ... Look up Edge in Wiktionary, the free dictionary. ... In mathematics, diagonal has a geometric meaning, and a derived meaning as used in square tables and matrix terminology. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ... A chord of a curve is a geometric line segment whose endpoints both lie on the curve. ...

Contents

Definition

If V,! is a vector space over mathbb{R} or mathbb{C}, and L,! is a subset of V,,! then L,! is a line segment if L,! can be parametrized as In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ...

L = { mathbf{u}+tmathbf{v} mid tin[0,1]}

for some vectors mathbf{u}, mathbf{v} in V,! with mathbf{v} neq mathbf{0}, in which case the vectors mathbf{u} and mathbf{u+v} are called the end points of L.,!


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 L,! that can be parametrized as

L = { mathbf{u}+tmathbf{v} mid tin(0,1)}

for some vectors mathbf{u}, mathbf{v} in V,! with mathbf{v} neq mathbf{0}.


An alternative, equivalent, definition is as follows: A (closed) line segment is a convex hull of two distinct points. In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ... A convex combination is a linear combination of data points (which can be vectors or scalars) where all coefficients are positive and sum up to 1. ...


Properties

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

In topology and related branches of mathematics, a connected space is a topological space which cannot be written as the disjoint union of two or more nonempty spaces. ... In set theory, a set is called non-empty (or nonempty) if it contains at least one element, and is therefore not the empty set. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...

See also

In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...

External links

  • Definiton of line segment With interactive animation
  • Copying a line segment with compass and straightedge Animated demonstration

This article incorporates material from Line segment on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative, online mathematics encyclopedia. ...


  Results from FactBites:
 
Line Segment (geometry) - Search Results - MSN Encarta (209 words)
Line Segment (geometry), portion of a line consisting of two points, called endpoints, and all of the points that lie between these endpoints.
The line is said to be tangent to the circle or is...
Chord (geometry), a line segment whose endpoints lie on a curve or on a circle.
  More results at FactBites »

 
 

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