nLab line segment

Idea

A line segment is a fundamental feature of geometry, going back to Euclid (and probably much earlier), and particularly relevant in convex spaces. In general, a line segment is that which lies between two points.

Definitions

In synthetic geometry, a line segment may be regarded as an unordered pair $\{p,q\}$ of points, called the endpoints of the line segment. A directed line segment is given by an ordered pair of points; so we speak of the directed line segment from $p$ to $q$ instead of the (undirected) line segment between $p$ and $q$. Generally, the endpoints are required to be distinct; otherwise the line segment is degenerate.

In analytic geometry, where geometric figures are conceived of as sets of points, the closed line segment between $p$ and $q$ consists of $p$, $q$, and every point in between, while the open line segment consists of the points strictly between $p$ and $q$ but not the endpoints themselves. (One can also consider half-open/half-closed versions.) Then a directed line segment from $p$ to $q$ (with the same open/closed variations) is a line segment between $p$ and $q$ equipped with an orientation.

In axiomatic approaches to geometry, the concept of ‘between’ from the previous paragraph is undefined, so that line segments are a fundamental concept. But in a convex space (including Cartesian spaces and other topological vector spaces), where we can form linear combinations of points (as long as the coefficients are positive and have unit sum) we may say that $r$ is between? $p$ and $q$ if $r = t p + (1 - t) q$ for some $t \in [0,1]$ and strictly between if this is so for some $t \in (0,1)$. Now the closed and open line segments are fully defined (in terms of the operation of taking convex-linear combinations). In other words, the closed line segment between $p$ and $q$ is the convex hull of the set $\{p,q\}$.

A line segment should be distinguished from a line (vastly generalized at line), which has no endpoints but continues to infinity. (There is also a ray, which has one endpoint but continues infinitely in the other direction.) In classical Euclidean geometry, lines are prominent, but they are treated as being only potentially infinite. So a line at any stage of construction is always a line segment, but one of the basic constructions is to extend such a line segment to a larger one, considered to still be the same line. In this way, a line can be thought of as an equivalence class of line segments, so equivalently as an equivalence class of pairs of (distinct) points.

Notation

The line segment between $p$ and $q$ is traditionally denoted $\overline{p q}$. The directed line segment from $p$ to $q$ may be denoted $\overrightarrow{p q}$, but this is also used for other things, including the ray from $p$ through $q$ (and then a double-headed arrow is used for the line through $p$ and $q$). Another way to write a directed line segment is $[p,q]$ (borrowing from notation for intervals), and sometimes this is even used for the undirected line segment, since it allows one to distinguish the closed line segment (as above) from the open line segment $(p,q)$.