--- categories: Physics --- *Snell's law* is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water, glass, or air. It is also of importance in some [optimization](Mathematical optimization) problems. For example, a lifeguard running to save a drowning person in the ocean will take a path that follows Snell's law. If the lifeguard can run at speed $v_1$ on the beach, and swim at speed $v_2$ in the ocean, then the angle $\theta_1$ at which the lifeguard enters the ocean (the angle of incidence) and the angle $\theta_2$ at which the lifeguard swims towards the drowning person (the angle of refraction) satisfies Snell's law if he takes the shortest possible path: $$ \frac{\sin\theta_1}{v_1} = \frac{\sin\theta_2}{v_2}. $$ > **Insight**: If Snell's law gives an undefined result (perhaps because of a domain error for $\sin^{-1}$), that means the light wouldn't be refracted, but reflected instead. In terms of optimization, this means that the given scenario is not optimal. ## Problems - [EllysThreeRivers](https://community.topcoder.com/stat?c=problem_statement&pm=11911&rd=14735) [^1] - [RemoteRover](https://community.topcoder.com/stat?c=problem_statement&pm=4022&rd=6534) [^2] - [Ironman](https://open.kattis.com/problems/ironman) ## External links - [Snell's law](https://en.wikipedia.org/wiki/Snell%27s_law) [^1]: [^2]: