If satisfies Huygens' construction, so that the secondary wavefront from is tangential to at , then is a path of stationary traversal time from to . Adding the fixed time from to , we find that is the path of stationary traversal time from to (possibly with a restricted domain of comparison, as noted above), in accordance with Fermat's principle. The argument works just as well in the converse direction, provided that has a well-defined tangent plane at . Thus Huygens' construction and Fermat's principle are geometrically equivalent.De Witte (1959) gives a more sophisticated proof of the same result, using calculus of variations.In Huygens' construction, the choice of the envelope of secondary wavefronts on the forward side of – that is, the rejection of "backward" or "retrograde" secondary waves – is also explained by Fermat's principle. For example, in Fig.2, the traversal time of the path (where the last leg "doubles back") is not stationary with respect to variation of , but is maximally sensitive to movement of along the leg .
if saTiSFIEs HuyGens' consTRUctiON, So ThaT the sEconDary wavefRONt frOM Is tAngENtIAL tO at , TheN is a paTh oF STatIonAry traveRSAl timE From tO . aDDinG tHE fixEd tIme from To , We fiNd That Is ThE Path of sTAtIonaRy TRaVersal TimE from To (PoSsiBlY witH a REstriCTed DomAin of compARISon, as NoTEd above), iN accordance WitH feRmaT's PriNciPLe. thE argUment woRks Just as WeLl in the ConVeRse DiRection, pRoVIdeD thAT Has a welL-defined tangEnt planE AT . Thus hUYgenS' ConstRUCtIon and FErMAt's pRIncIPle ARE GeometricAlly EquiValent.DE witte (1959) gives a more sophIsticateD pRooF oF the SAme reSULt, usIng Calculus of VArIatIonS.in HUYgEnS' conStruction, tHe ChoicE of tHe enVelopE Of sEcOndaRy wavefronts oN thE FORwArd side of – thAt is, tHe RejECtion of "backWard" or "retROgRADe" secondary waves – is also ExplAineD bY fermat's prinCiPle. for EXamplE, IN fIG.2, the travErsal timE of the pAth (wherE ThE LAst leg "doUBlEs back") is not sTationary WITH reSpect TO variatIon of , But Is maXimalLy sensITive tO movemeNT oF AlOnG tHe lEg .