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Concept
Line segment
Summary
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. 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 a line above the symbols for the two endpoints (such as ).
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).
If V is a vector space over \R or \C, 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 added to the distance is equal to the distance . Thus in \R^2, the line segment with endpoints and is the following collection of points:
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.
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