Oriented angle

Definition

Let $u = (u_1, u_2)$ and $v = (v_1, v_2)$ be nonzero vectors in the Euclidean plane $\mathbb{R}^2$. Let $u \cdot v = u_1 v_1 + u_2 v_2$ and $\det(u, v) = u_1 v_2 - u_2 v_1$. The oriented angle from $u$ to $v$ is the class modulo $2\pi$ of each real number $\theta$ with $$\cos \theta = \frac{u \cdot v}{\|u\| \, \|v\|} \quad \text{and} \quad \sin \theta = \frac{\det(u, v)}{\|u\| \, \|v\|}.$$ For points $A$, $B$ and $C$ of $\mathbb{R}^2$ with $A \neq B$ and $C \neq B$, the oriented angle $\measuredangle ABC$ is the oriented angle from $A - B$ to $C - B$.

Remarks

Such a $\theta$ exists because $(u \cdot v)^2 + \det(u, v)^2 = \|u\|^2 \|v\|^2$. It is unique modulo $2\pi$. Swapping the vectors changes the sign: the oriented angle from $v$ to $u$ is minus the oriented angle from $u$ to $v$. The unoriented angle between $u$ and $v$ is the absolute value of the representative of the oriented angle in $(-\pi, \pi]$. Mathlib's EuclideanGeometry.oangle works in each oriented Euclidean plane, with values in $\mathbb{R} / 2\pi\mathbb{Z}$. It gives $0$ when a vector is zero.

Explanations

L1 human

The oriented angle remembers the direction of the turn. Turning from $(1, 0)$ to $(0, 1)$ is a quarter turn anticlockwise, so the oriented angle is $\pi / 2$. Turning back, from $(0, 1)$ to $(1, 0)$, gives $-\pi / 2$.