Definitions

open-line

Line segment

In geometry, a line segment is a part of a line that is bounded by two distinct end points, 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, 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).

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 parameterized 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}, 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.

Properties

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