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Angle of Incidence

The angle between the incident ray and the normal to the surface at the point of incidence, used in reflection and refraction calculations.
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The statement of the theorem

Let Σ⊂R3\Sigma \subset \mathbb{R}^3 be the reflecting surface, and let P∈ΣP \in \Sigma be the point of incidence. Define n^∈R3\mathbf{\hat{n}} \in \mathbb{R}^3 as the unit normal vector to Σ\Sigma at PP. Let k^i∈R3\mathbf{\hat{k}}_i \in \mathbb{R}^3 be the unit vector representing the direction of the incident wave propagation (the incident ray). The angle of incidence, θi\theta_i, is defined by the relationship:\n\ncos⁡(θi)=∣k^i⋅n^∣\cos(\theta_i) = |\mathbf{\hat{k}}_i \cdot \mathbf{\hat{n}}| \n\nEquivalently, θi\theta_i is the unique angle in the interval [0,π/2][0, \pi/2] such that:\n\nθi=arccos⁡(∣k^i⋅n^∣)\theta_i = \arccos\left( |\mathbf{\hat{k}}_i \cdot \mathbf{\hat{n}}| \right)