The visual intuition through which we represent the Euclidean plane suggests that we can orient it according to a direction of rotation. This intuition reflects a rigorous mathematical definition of the orientation of the plane, which involves choosing a basis and, implicitly, an angle between vectors. In fact, choosing an orientation corresponds to selecting a family of orthonormal bases that are related to each other by a direct isometry. Alternatively, it can be considered as the selection of a family of arbitrary bases that are related to each other by a transformation with a strictly positive determinant.

1. Orienting the Plane: Intuition, Representation, and Basis Vectors

1.1. Two Possible Directions of Rotation in the Plane

Let us choose a point in the Euclidean plane and draw two half-lines in different directions starting from this point. Intuition suggests that we can “move” from one half-line to the other by two different rotations, depending on the chosen direction. The “direct” or “trigonometric” direction, according to our usual representation of the plane, corresponds to the “counterclockwise” direction. The “indirect” direction corresponds to the other direction. This idea, that two possible directions of rotation exist in the plane, corresponds to a rigorous mathematical concept: the orientation of the Euclidean plane.

1.2. The Visual Representation of Orientation Is a Convention

The Euclidean plane (that is, the intuition underlying Euclidean geometry) is represented in modern mathematics by the Cartesian product $\mathbb{R}^2$. The pair of vectors $\vec{i} = (1,0)$ and $\vec{j} = (0,1)$, called the canonical basis, is traditionally represented as shown in the figure below. Such a representation establishes an “orientation” in our minds: the “smallest angle” (the angle with the smallest principal measure) between the two vectors $\vec{i}$ and $\vec{j}$ determines the “trigonometric” direction. It is important to remember that this is merely a choice of representation, based on the increasing order of abscissas from left to right! If we were to represent the abscissas in the opposite direction, then our representation of the orientation determined by $\vec{i}$ and $\vec{j}$ would be reversed. Similarly, we could represent the abscissas vertically and the ordinates horizontally. This means that there is no inherently privileged visual representation of this “standard” orientation.

1.3. The Order of the Basis Vectors Defines an Orientation

However, regardless of our visual representation, the choice of $\vec{i}$ and $\vec{j}$, in that specific order, determines a privileged orientation in a precise mathematical sense. In other words, the orientation is determined by the order $\vec{i}, \vec{j}$, which means that taking the two vectors in the reverse order, $\vec{j}, \vec{i}$, defines a different orientation. Since only two possible orientations of the plane exist (as we shall see), the chosen order of these two vectors constitutes a possible definition of a plane’s orientation.

2. Mathematical Definition of Plane Orientation

2.1. Orienting the Euclidean Plane by Choosing a Basis

We aim to give a precise mathematical meaning to the idea that the two (different) pairs formed by the two vectors $\vec{i}$ and $\vec{j}$ of the canonical basis of $\mathbb{R}^2$ determine the two (and only two) possible orientations of the plane. Recall that this pair of vectors is called a “basis of the vector space $\mathbb{R}^2$” because they allow any vector $\vec{u} = (x, y)\in (\mathbb{R}^2$ to be uniquely represented in the form $\vec{u} = x.\vec{i} + y.\vec{j}$. The essential reason for this is that these two vectors are nonzero and non-collinear (i.e., they have different directions). It is evident that the choice of a different basis also determines an orientation: two vectors of a basis determine an oriented angle, neither null nor flat, and the principal measure of this angle, whether less than or greater than $\pi$, determines an orientation, respectively “direct” or “indirect” (with respect to the canonical basis, taken as a reference). We must, therefore, precisely establish the relationship between plane bases and orientation, and how these notions relate to the vectors $\vec{i}$ and $\vec{j}$.

2.2. The Angle Defined by a Plane Basis

To any basis $B = (\vec{u}, \vec{v})$ of the Euclidean plane, we can associate a vector angle. By dividing $\vec{u}$ and $\vec{v}$ by their norms, we obtain a pair of unit vectors $B’ = (\vec{u’}, \vec{v’})$, which by definition determine such an angle $\vec{\alpha} = [(\vec{u’}, \vec{v’})]$; let us call it the angle determined by $B$. Since $\vec{u}$ and $\vec{u’}$ on one hand, and $\vec{v}$ and $\vec{v’}$ on the other, share the same direction, $B’$ is also a basis of $\mathbb{R}^2$. In other words, the vectors $\vec{u’}$ and $\vec{v’}$ are nonzero and non-collinear, so the principal measure of the angle determined by $B$, i.e., the number $s \in [0, 2\pi[$ representing this measure, is neither $0$ nor $\pi$. The value of $s$ is then either in the open interval $]0, \pi[$ or in the open interval $]\pi, 2\pi[$. The intuition underlying the idea of a basis $B$ oriented “in the direct sense” corresponds to the case where the principal measure $s$ of the angle $\vec{\alpha}$ is the smallest possible, i.e., $0 < s < \pi$. Otherwise, i.e., when $\pi < s < 2\pi$, the basis $B$ is considered to be oriented “in the indirect sense.” Naturally, swapping the vectors $\vec{u}$ and $\vec{v}$ “reverses the orientation,” since the angle $\vec{\alpha}$ is replaced by $-\vec{\alpha}$, whose principal measure is $2\pi – s$.

2.3. Algebraic Interpretation of Basis Orientation

Choosing a plane orientation can thus be understood as choosing a basis of the plane. The principal measure of the angle defined by the vectors of this basis determines by definition whether the orientation is direct or indirect. The canonical basis provides a direct orientation since the angle $[(\vec{i}, \vec{j})]$ has a principal measure of $\pi/2$, which lies in the interval $]0, \pi[)$ It is, in fact, possible to characterize the orientation determined by any given basis by relating it to the canonical basis. Given a basis $B = (\vec{u}, \vec{v})$, we previously discussed the existence of a linear transformation that allows one to transition from the canonical basis to the basis $B$: this is the mapping $f : (x, y) \in \mathbb{R}^2 \mapsto (ax + by, cx + dy)$, if $\vec{u} = (a, c)$ and $\vec{v} = (b, d).$ Generally, we have mentioned how the determinant of such a transformation $f$, i.e., the number $\Delta = ad – bc$, characterizes the situations where the vectors $(a, c)$ and $(b, d)$ form a basis, namely when $ad – bc \neq 0$. In this case, the determinant also determines the orientation of the basis depending on its sign: it can be shown that the basis $B$ is direct when $\Delta > 0$, and indirect when $\Delta < 0$.

3. Orientation Using Orthonormal Bases

3.1. Characterizing the Orientation Induced by an Orthonormal Basis

When introducing orthonormal bases, we specifically distinguished between direct orthonormal bases $(\vec{u}, \vec{v})$, for which the angle $\vec{\alpha} = [(\vec{u}, \vec{v})]$ has a principal measure of $\pi/2$, and indirect orthonormal bases $(\vec{u}, \vec{v})$, for which the angle $\vec{\alpha} = [(\vec{u}, \vec{v})]$ has a measure of $-\pi/2$. This corresponds to a particular case of the orientation defined here using a basis, as an orthonormal basis is indirect when its principal measure is $2\pi – \pi/2 = 3\pi/2$, within the interval $]\pi, 2\pi[$. The interpretation of orientation via the determinant is particularly simple in this case: if $B = (\vec{u}, \vec{v})$ is an orthonormal basis and $f$ is its associated orthogonal transformation, the basis $B$ is direct when the determinant $\Delta$ of $f$ equals $1$, and indirect when $\Delta$ equals $-1$. This straightforward characterization of plane orientation often leads to defining it based on orthonormal bases, specifically equivalence classes of such bases.

3.2. Equivalent Orthonormal Bases: Two Possible Outcomes

Given any two orthonormal bases $B$ and $B’$, the transformation $f$ from $B$ to $B’$ is always a vector isometry, and only two cases can occur. Either $f$ is direct (i.e., it has a determinant of $1$), or $f$ is indirect (i.e., it has a determinant of $-1$). In the first case, where $f$ is a rotation, we say that $B$ and $B’$ are equivalent. An equivalence class of orthonormal bases is then simply a set of bases equivalent to a given one. For any given orthonormal basis $B$, the transformation from the canonical basis $C = (\vec{i}, \vec{j})$ to $B$ is either direct or indirect, and these are the only two possibilities! Thus, there are only two equivalence classes: the set of orthonormal bases derived from $C$ by rotation, and the set of orthonormal bases derived from $C$ by reflection (a transformation with determinant $-1$). Plane orientation is formally defined as the choice of one of these classes, which then serves as a reference.

3.3. Defining Orientation Using the Equivalence of Arbitrary Bases

This approach based on orthonormal bases is also applicable to any type of basis. If $B$ and $B’$ are two arbitrary bases of the plane, the linear transformation $f$ from $B$ to $B’$ is generally not an isometry but always has a nonzero determinant $\Delta$. Here too, two cases naturally arise: either $\Delta > 0$ (the transformation $f$ is “direct” or “preserves orientation”), or $\Delta < 0$ (the transformation $f$ is “indirect” or “reverses orientation”). We can also say that $B$ and $B’$ are equivalent if $\Delta > 0$, and by again taking the canonical basis $C = (\vec{i}, \vec{j})$ as a reference, distinguish between bases derived from it by direct transformations and those derived from it by indirect transformations. This again yields two equivalence classes, and choosing one of them also determines a plane orientation.

Conclusion: Plane Orientation Remains a Mathematical Choice

There are several mathematically rigorous ways to define the orientation of the Euclidean plane ($\mathbb{R}^2$). While there is a natural privileged orientation given by the canonical basis and other bases, orthonormal or not, equivalent to it via a direct transformation, orienting the plane ultimately results from a choice, either of a basis or of an equivalence class of plane bases. However, there are always only two possible choices.