Oriented angle

Builds on Euclidean plane Angle between two vectors

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.

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Tag	`0G4W89YJRK`
URI	`m2://0G4W89YJRK`
Kind	definition
Writing	W0 stub
MSC	51M05
Defines	Oriented angle
Uses	Euclidean plane, Angle between two vectors

Oriented angle

Definition

Remarks

Reading path

Further reading

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