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line segment∗
matte†
2013-03-21 17:20:11
Definition Suppose 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
L = {a + tb | t ∈ [0, 1]}
for some a, b in V with b 6= 0.
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 = {a + tb | t ∈ (0, 1)}
for some a, b in V with b 6= 0.
If x and y are two vectors in V and x 6= y, then we denote by [x, y] the set
connecting x and y. This is , {αx + (1 − α)y |0 ≤ α ≤ 1}. One can easily check
that [x, y] is a closed line segment.
Remarks
• An alternative, equivalent, definition is as follows: A (closed) line segment
is a convex hull of two distinct points.
• A line segment is 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.
∗ hLineSegmenti
created: h2013-03-21i by: hmattei version: h35783i Privacy setting: h1i
hDefinitioni h03-00i h51-00i
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