Line segment explained

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. 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

is a vector space over or , and is a subset of then is a line segment if can be parameterized as

for some vectors

, in which case the vectors and are called the end points of

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

that can be parametrized as

for some vectors

.

An alternative, equivalent, definition is as follows: A (closed) line segment is a convex hull of two points.

Properties

• A line segment is a connected, non-empty set.
• If

is a topological vector space, then a closed line segment is a closed set in However, an open line segment is an open set in if and only if is one-dimensional.

• More generally than above, the concept of a line segment can be defined in an ordered geometry.

In Proofs

In geometry, the segment addition postulate states that if B is between A and C, then segment AB + segment BC = segment AC.

See also

• Interval (mathematics)
• Line (geometry)

References

• David Hilbert: The Foundations of Geometry. The Open Court Publishing Company 1950, p. 4

External links

• Line Segment at PlanetMath]]
• Definition of line segment With interactive animation
• Copying a line segment with compass and straightedge
• Dividing a line segment into N equal parts with compass and straightedge Animated demonstration

----