Fermat’s principle of least time

E141903

optical principle physical law variational principle

Fermat’s principle of least time is a fundamental variational principle in optics stating that light follows the path that takes the least time, from which many laws of geometrical optics can be derived.

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All labels observed (5)

Label	Occurrences
Fermat's principle	2
Fermat’s principle	1
Fermat’s principle in optics	1
Fermat’s principle of least time canonical	1
Hamilton's optico-mechanical analogy	1

How this entity was disambiguated

This entity first appeared as the object of triple T1249088 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Fermat’s principle of least time
Context triple: [Snell’s law of refraction, derivedFrom, Fermat’s principle of least time]

A. Huygens–Fresnel principle
The Huygens–Fresnel principle is a fundamental concept in wave optics that explains how every point on a wavefront acts as a source of secondary wavelets whose interference determines the wave’s subsequent propagation and diffraction.
B. Principles of Optics
Principles of Optics is a seminal textbook that rigorously develops the theory of electromagnetic waves and optical phenomena, profoundly shaping modern physical optics.
C. Kirchhoff diffraction theory
Kirchhoff diffraction theory is a classical wave optics framework that models light propagation and diffraction by treating wavefronts as superpositions of secondary spherical waves emitted from an aperture.
D. Snell’s law of refraction
Snell’s law of refraction is a fundamental principle in optics that relates the angles of incidence and refraction to the refractive indices of two media, governing how light bends when passing between them.
E. Traité de la lumière
Traité de la lumière is a seminal 1690 scientific treatise that presents Christiaan Huygens’ wave theory of light, including the principle now known as Huygens’ principle.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Fermat’s principle of least time
Target entity description: Fermat’s principle of least time is a fundamental variational principle in optics stating that light follows the path that takes the least time, from which many laws of geometrical optics can be derived.

A. Huygens–Fresnel principle
The Huygens–Fresnel principle is a fundamental concept in wave optics that explains how every point on a wavefront acts as a source of secondary wavelets whose interference determines the wave’s subsequent propagation and diffraction.
B. Principles of Optics
Principles of Optics is a seminal textbook that rigorously develops the theory of electromagnetic waves and optical phenomena, profoundly shaping modern physical optics.
C. Kirchhoff diffraction theory
Kirchhoff diffraction theory is a classical wave optics framework that models light propagation and diffraction by treating wavefronts as superpositions of secondary spherical waves emitted from an aperture.
D. Snell’s law of refraction
Snell’s law of refraction is a fundamental principle in optics that relates the angles of incidence and refraction to the refractive indices of two media, governing how light bends when passing between them.
E. Traité de la lumière
Traité de la lumière is a seminal 1690 scientific treatise that presents Christiaan Huygens’ wave theory of light, including the principle now known as Huygens’ principle.
F. None of above. chosen

Statements (45)

Predicate	Object
instanceOf	optical principle ⓘ physical law ⓘ variational principle ⓘ
alsoKnownAs	Fermat’s principle of least time ⓘ surface form: Fermat’s principle principle of least time ⓘ
appliesTo	light propagation in media with varying refractive index ⓘ optical path in graded-index media ⓘ reflection of light at interfaces ⓘ refraction of light at interfaces ⓘ
approximationType	short-wavelength limit of wave optics ⓘ
assumes	light speed depends on medium refractive index ⓘ light travels locally in straight lines in homogeneous media ⓘ
clarification	least time means stationary time, not necessarily global minimum ⓘ
compatibleWith	quantum electrodynamics path integral viewpoint ⓘ wave theory of light ⓘ
coreIdea	actual path of a light ray makes the optical path length stationary with respect to nearby paths ⓘ light follows the path that requires the least time between two points ⓘ
didNotOriginallyUse	calculus of variations formalism ⓘ
explains	equal-time paths in reflection from a plane mirror ⓘ multiple optical paths in phenomena like mirages ⓘ path bending in media with spatially varying refractive index ⓘ
field	geometrical optics ⓘ optics ⓘ
framework	calculus of variations ⓘ
generalizationOf	straight-line propagation of light in uniform media ⓘ
historicalPeriod	17th century ⓘ
inspired	development of variational methods in physics ⓘ
involvesQuantity	optical path length ⓘ refractive index ⓘ travel time of light ⓘ
mathematicalFormulation	optical path length is stationary under small variations of the path ⓘ δ∫n(s) ds = 0 for the actual light path ⓘ
namedAfter	Pierre de Fermat ⓘ
refines	principle of least action in optics ⓘ
relatedTo	Hamiltonian optics ⓘ Huygens–Fresnel principle ⓘ surface form: Huygens’ principle principle of stationary action ⓘ
status	foundational principle in classical optics ⓘ
usedFor	analysis of optical paths in lenses ⓘ derivation of Snell’s law of refraction ⓘ derivation of eikonal equation ⓘ derivation of laws of geometrical optics ⓘ derivation of the law of reflection ⓘ ray tracing in inhomogeneous media ⓘ
validInLimit	geometrical optics approximation ⓘ

How these facts were elicited

Referenced by (6)

Full triples — surface form annotated when it differs from this entity's canonical label.

Snell’s law of refraction → derivedFrom → Fermat’s principle of least time ⓘ

Pierre de Fermat → notableWork → Fermat’s principle of least time ⓘ

this entity surface form: Fermat's principle

Euler–Lagrange equation → generalizes → Fermat’s principle of least time ⓘ

this entity surface form: Fermat’s principle in optics

William Rowan Hamilton → knownFor → Fermat’s principle of least time ⓘ

this entity surface form: Hamilton's optico-mechanical analogy

Fermat’s principle of least time → alsoKnownAs → Fermat’s principle of least time ⓘ

this entity surface form: Fermat’s principle

brachistochrone problem → relatedConcept → Fermat’s principle of least time ⓘ