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Parallel Rays

Light originates as parallel rays, a simplifying assumption that allows for the application of geometrical optics techniques.
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The statement of the theorem

Let r:R→R3\mathbf{r}: \mathbb{R} \to \mathbb{R}^3 be the position vector describing a ray path, parameterized by t∈Rt \in \mathbb{R}. Define the direction vector k^∈S2\hat{\mathbf{k}} \in \mathbb{S}^2 as a fixed unit vector. The condition for r(t)\mathbf{r}(t) to represent a set of parallel rays is that the tangent vector r′(t)\mathbf{r}'(t) must be constant and equal to k^\hat{\mathbf{k}} for all tt. Formally, this requires the ray path to satisfy the differential equation:\ndrdt=k^\frac{d\mathbf{r}}{dt} = \hat{\mathbf{k}} \nIntegrating this yields the parametric form of the ray: r(t)=r0+tk^\mathbf{r}(t) = \mathbf{r}_0 + t \hat{\mathbf{k}}, where r0=r(0)\mathbf{r}_0 = \mathbf{r}(0) is the initial position vector. The set of all such rays R\mathcal{R} is thus defined by the family of curves rr0,k^(t)=r0+tk^\mathbf{r}_{\mathbf{r}_0, \hat{\mathbf{k}}}(t) = \mathbf{r}_0 + t \hat{\mathbf{k}}, parameterized by the initial point r0∈R3\mathbf{r}_0 \in \mathbb{R}^3 and the fixed direction k^\hat{\mathbf{k}}.