N-Dimensional Line Segments

Definition

The closed line segment (or interval) between two points in $n$-dimensional space is the set of points which can be expressed as the sum of the first point and a scalar multiple of the difference between the second point and the first; where the scalar is in the interval $[0, 1]$. Thus, the closed line segment between two points is a subset of the line though the two points. The open line segment between $x$ and $y$ is the closed line segment with the points $x$ and $y$.

Notation

We denote the closed line segment between $x$ and $y$ by $[x, y]$. So,

\[ [x,y] = \Set*{x + \alpha (y-x)}{0 \leq \alpha \leq 1} \]

Notice that $x + \alpha (y - x) = (1-\alpha )x + \alpha y$. Similarly, we denote the open line segment by $(x, y)$. These notations pleasantly generalize that of real intervals.