Editing Envelope (mathematics) (section)

===Huygens's principle===
Light may pass through anisotropic inhomogeneous media at different rates depending on the direction and starting position of a light ray.  The boundary of the set of points to which light can travel from a given point '''q''' after a time ''t'' is known as the [[wave front]] after time ''t'', denoted here by Φ<sub>'''q'''</sub>(''t'').  It consists of precisely the points that can be reached from '''q''' in time ''t'' by travelling at the speed of light.  [[Huygens–Fresnel principle|Huygens's principle]] asserts that the wave front set {{nowrap|&Phi;<sub>'''q'''<sub>0</sub></sub>(''s'' + ''t'')}} is the envelope of the family of wave fronts {{nowrap|&Phi;<sub>'''q'''</sub>(''s'')}} for '''q'''&nbsp;∈&nbsp;Φ<sub>'''q'''<sub>0</sub></sub>(''t''). More generally, the point '''q'''<sub>0</sub> could be replaced by any curve, surface or closed set in space.<ref>{{Citation | last1=Arnold | first1=V. I. | author1-link=Vladimir Arnold | title=Mathematical Methods of Classical Mechanics, 2nd ed. | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-96890-2 | year=1997 | url-access=registration | url=https://archive.org/details/mathematicalmeth0000arno }}, §46.</ref>