Fermat's principle

Fermat's principle states that light follows paths between points for which travel time is stationary under small variations. It underlies geometrical optics, including reflection and refraction.

Principle of least action: A variational principle stating that physical trajectories make an action functional stationary. Fermat's principle is its optical counterpart, with travel time playing the role of action.
Refractive index: A dimensionless measure of how much slower light propagates in a medium than in vacuum. It determines the local travel time that Fermat's principle compares between paths.
Pierre de Fermat: A seventeenth-century French mathematician who made foundational contributions to number theory, probability, and optics. Fermat proposed that light's path could be characterized by least travel time.
Huygens–Fresnel principle: A wave construction in which each point on a wavefront acts as a source of secondary wavelets. It describes light through wave propagation rather than selecting a single stationary-time ray.
Euler–Lagrange equation: A differential equation giving extrema or stationary points of functionals. Applying this variational equation to optical travel time yields ray paths.
Speed of light: The speed at which light propagates, equal to a fixed value in vacuum and dependent on medium in materials. Travel time along each candidate path depends on light's speed through the traversed media.
Heron of Alexandria: An ancient Greek mathematician and engineer associated with an early geometric treatment of reflection. Heron's shortest-path argument for reflection foreshadowed a variational approach to optics.
Diffraction: The spreading and interference of waves as they pass obstacles or openings. Diffraction produces behavior that geometrical rays and stationary paths alone cannot explain.
Snell's law: The relation between the angles of incidence and refraction at an interface between media. Stationary travel time across an interface produces this relation between ray angles.
Ray (optics): An idealized line indicating the direction in which light energy propagates. Fermat's principle determines the paths represented by optical rays.

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