line segment (original) (raw)

DefinitionSuppose 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 parametrized as

for some a,b in V with b≠0.

Sometimes one needs to distinguish between open and closed (http://planetmath.org/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

for some a,b in V with b≠0.

If x and y are two vectors in V and x≠y, then we denote by[x,y] the set connecting x and y. This is , {α⁢x+(1-α)⁢y|0≤α≤1}. One can easily check that [x,y] is a closed line segment.

Remarks

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  An alternative, equivalent, definition is as follows: A (closed) line segment is a convex hull of two distinct points.
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  A line segment is connected, non-empty set.
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  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.
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  More generally than above, the concept of a line segment can be defined in an ordered geometry.