Rigorous definition of "oriented line" in an Euclidean affine space

Question

Let $\mathcal{A}^n$ be an affine space of dimension $n$. For example, let's take $n=3$.

A line $\mathcal{s}$ of $\mathcal{A}^3$ is an affine subspace of dimension $1$, that is:

$\mathcal{s}=\{P \in \mathcal{A}_3 \text{ such that } \overrightarrow{AP} \in \langle u \rangle \}$.

Now, what is not clear to me is:

if we consider an Euclidean affine space $\mathcal{E}^3$, what is the (rigorous) definition of an oriented line?

Answer 1

First one has to define line segments and prove a lemma.

You might already have your favorite definition of a line segment in an affine space, here's just one possibility: For any $a \ne b$ in the affine space define the segment $[a,b]$ to be the set of points $p$ for which there exists $t \in [0,1]$ such that $p = a + t \cdot \overrightarrow{ab}$.

Lemma: For each line $s$ there exist exactly two total orders $\le$ such that for each $a<b$ in $s$ we have $[a,b] = \{x \in s \bigm| a \le x \le b\}$.

Once that lemma has been proved, an orientation of $s$ is a choice of one of those two total orders.

Answer 2

Informally speaking, an oriented line of an affine space is a one-dimensional affine subspace together with an orientation selected for its direction.

Definition 1. Suppose $L$ is a one-dimensional vector space over $\mathbb{R}$. An orientation on $L$ is a choice of $\mathbb{R}^+ v = \{\lambda v\ |\ \lambda\in\mathbb{R}^+\}$ or $\mathbb{R}^- v= \{\lambda v\ |\ \lambda\in\mathbb{R}^-\}$ where $v\neq 0, v\in L$.

The sets $\mathbb{R}^+ v$ and $\mathbb{R}^- v$ are two equivalence classes among bases of $L$ that have the same orientation.

Suppose $V$ is a vector space over $\mathbb{R}$ and $A$ is an affine space with tangent space $V$.

Definition 2. An oriented line $\ell$ of $A$ is an affine subspace $a + L$ where $a$ is a point of $A$ and $L$ a one-dimensional subspace of $V$, together with an orientation of $L$.

Definition 3. Suppose $a, b \in A$ are two points $a \neq b$. An oriented line from $a$ to $b$, denoted by $a\vee b$, is an affine subspace $\{(a + \lambda (b-a)\ |\ \lambda \in \mathbb{R}\}$ with the orientation $\{\mathbb{R}^+(b-a)\}$ and an opposite oriented line $-(a\vee b)$ is defined an oriented line from $b$ to $a$, i.e.

\begin{align}a\vee b &= \{(a + \lambda (b-a)\ |\ \lambda \in \mathbb{R}\}\cup\{\mathbb{R}^+(b-a)\}, \\ b\vee a & = \{(b + \lambda (a-b)\ |\ \lambda \in \mathbb{R}\}\cup\{\mathbb{R}^+(a-b)\} = \\ & = \{(a + \lambda (b-a)\ |\ \lambda \in \mathbb{R}\}\cup\{\mathbb{R}^-(b-a)\} = -(a\vee b).\end{align}

Answer 3

The simplest definition would be the pair of an affine line and an orthonormal basis of its tangent space (which is a Euclidean vector space of dimension$~1$). In higher dimensions one would need to choose a connected component of the set of orthonormal bases to choose an orientation, but in dimension$~1$ there are only two orthonormal bases (of one vector each), so it is just the choice of one of them.