Editing Envelope (mathematics) (section)

==Applications==

=== Ordinary differential equations ===
Envelopes are connected to the study of [[ordinary differential equation]]s (ODEs), and in particular [[singular solution]]s of ODEs.<ref>{{Citation | last1=Forsyth | first1=Andrew Russell | title=Theory of differential equations | publisher=[[Dover Publications]] | location=New York | series=Six volumes bound as three | mr=0123757 | year=1959}}, §§100-106.</ref> Consider, for example, the one-parameter family of tangent lines to the parabola ''y'' = ''x''<sup>2</sup>. These are given by the generating family {{nowrap|1=''F''(''t'',(''x'',''y'')) = ''t''<sup>2</sup> – 2''tx'' + ''y''}}. The zero level set {{nowrap|1=''F''(''t''<sub>0</sub>,(''x'',''y'')) = 0}} gives the equation of the tangent line to the parabola at the point (''t''<sub>0</sub>,''t''<sub>0</sub><sup>2</sup>). The equation {{nowrap|1=''t''<sup>2</sup> – 2''tx'' + ''y'' = 0}} can always be solved for ''y'' as a function of ''x'' and so, consider
:<math> t^2 - 2tx + y(x) = 0. \ </math>
Substituting
:<math> t = \left(\frac{dy}{dx}\right)/2 </math>
gives the ODE
:<math> \left(\frac{dy}{dx}\right)^2 \!\! - 4x\frac{dy}{dx} + 4y = 0. </math>
Not surprisingly ''y''&nbsp;=&nbsp;2''tx''&nbsp;&minus;&nbsp;''t''<sup>2</sup> are all solutions to this ODE. However, the envelope of this one-parameter family of lines, which is the parabola ''y''&nbsp;=&nbsp;''x''<sup>2</sup>, is also a solution to this ODE. Another famous example is [[Clairaut's equation]].

=== Partial differential equations ===
Envelopes can be used to construct more complicated solutions of first order [[partial differential equation]]s (PDEs) from simpler ones.<ref>{{Citation | last1=Evans | first1=Lawrence C. | title=Partial differential equations | publisher=[[American Mathematical Society]] | location=Providence, R.I. | isbn=978-0-8218-0772-9 | year=1998}}.</ref>  Let ''F''(''x'',''u'',D''u'')&nbsp;=&nbsp;0 be a first order PDE, where ''x'' is a variable with values in an open set Ω&nbsp;⊂&nbsp;'''R'''<sup>''n''</sup>, ''u'' is an unknown real-valued function, D''u'' is the [[gradient]] of ''u'', and ''F'' is a continuously differentiable function that is regular in D''u''.  Suppose that ''u''(''x'';''a'') is an ''m''-parameter family of solutions:  that is, for each fixed ''a''&nbsp;∈&nbsp;''A''&nbsp;⊂&nbsp;'''R'''<sup>''m''</sup>, ''u''(''x'';''a'') is a solution of the differential equation.  A new solution of the differential equation can be constructed by first solving (if possible)
:<math>D_a u(x;a) = 0\,</math>
for ''a''&nbsp;=&nbsp;φ(''x'') as a function of ''x''.  The envelope of the family of functions {''u''(·,''a'')}<sub>''a''∈''A''</sub> is defined by
:<math>v(x) = u(x;\varphi(x)),\quad x\in\Omega,</math>
and also solves the differential equation (provided that it exists as a continuously differentiable function).

Geometrically, the graph of ''v''(''x'') is everywhere tangent to the graph of some member of the family ''u''(''x'';''a''). Since the differential equation is first order, it only puts a condition on the tangent plane to the graph, so that any function everywhere tangent to a solution must also be a solution. The same idea underlies the solution of a first order equation as an integral of the [[Monge cone]].<ref>{{citation |first=Fritz |last=John |authorlink=Fritz John |title=Partial differential equations |publisher=Springer |edition=4th |year=1991 |isbn=978-0-387-90609-6 |url-access=registration |url=https://archive.org/details/partialdifferent00john_0 }}.</ref>  The Monge cone is a cone field in the '''R'''<sup>''n''+1</sup> of the (''x'',''u'') variables cut out by the envelope of the tangent spaces to the first order PDE at each point. A solution of the PDE is then an envelope of the cone field.

In [[Riemannian geometry]], if a smooth family of [[geodesic]]s through a point ''P'' in a [[Riemannian manifold]] has an envelope, then ''P'' has a [[conjugate point]] where any geodesic of the family intersects the envelope. The same is true more generally in the [[calculus of variations]]: if a family of extremals to a functional through a given point ''P'' has an envelope, then a point where an extremal intersects the envelope is a conjugate point to ''P''.

=== Caustics ===
[[Image:Circle caustic.png|thumb|Reflective caustic generated from a [[circle]] and parallel rays]]
In [[geometrical optics]], a [[Caustic (optics)|caustic]] is the envelope of a family of [[ray (optics)|light rays]]. In this picture there is an [[circular arc|arc]] of a circle. The light rays (shown in blue) are coming from a source ''at infinity'', and so arrive parallel. When they hit the circular arc the light rays are scattered in different directions according to the [[Specular reflection|law of reflection]]. When a light ray hits the arc at a point the light will be reflected as though it had been reflected by the arc's [[tangent line]] at that point. The reflected light rays give a one-parameter family of lines in the plane. The envelope of these lines is the [[Caustic (optics)|reflective caustic]]. A reflective caustic will generically consist of [[smooth curve|smooth]] points and [[cusp (singularity)|ordinary cusp]] points.

From the point of view of the calculus of variations, [[Fermat's principle]] (in its modern form) implies that light rays are the extremals for the length functional
:<math>L[\gamma] = \int_a^b |\gamma'(t)|\,dt</math>
among smooth curves γ on [''a'',''b''] with fixed endpoints γ(''a'') and γ(''b'').  The caustic determined by a given point ''P'' (in the image the point is at infinity) is the set of conjugate points to ''P''.<ref>{{Citation | last1=Born | first1=Max | author1-link=Max Born | title=Principle of Optics | publisher=[[Cambridge University Press]] | isbn=978-0-521-64222-4 | date=October 1999 }}, Appendix I: The calculus of variations.</ref>

===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>