Caustic (mathematics)
Template:Short description {{#invoke:other uses|otheruses}}
In differential geometry, a caustic is the envelope of rays either reflected or refracted by a manifold. It is related to the concept of caustics in geometric optics. The ray's source may be a point (called the radiant) or parallel rays from a point at infinity, in which case a direction vector of the rays must be specified.
More generally, especially as applied to symplectic geometry and singularity theory, a caustic is the critical value set of a Lagrangian mapping Template:Nowrap where Template:Nowrap is a Lagrangian immersion of a Lagrangian submanifold L into a symplectic manifold M, and Template:Nowrap is a Lagrangian fibration of the symplectic manifold M. The caustic is a subset of the Lagrangian fibration's base space B.<ref name="Arnold">Template:Cite book</ref>
ExplanationEdit
Concentration of light, especially sunlight, can burn. The word caustic, in fact, comes from the Greek καυστός, burnt, via the Latin causticus, burning.
A common situation where caustics are visible is when light shines on a drinking glass. The glass casts a shadow, but also produces a curved region of bright light. In ideal circumstances (including perfectly parallel rays, as if from a point source at infinity), a nephroid-shaped patch of light can be produced.<ref>Circle Catacaustic. Wolfram MathWorld. Retrieved 2009-07-17.</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Rippling caustics are commonly formed when light shines through waves on a body of water.
Another familiar caustic is the rainbow.<ref>Rainbow caustics</ref><ref>Caustic fringes</ref> Scattering of light by raindrops causes different wavelengths of light to be refracted into arcs of differing radius, producing the bow.
CatacausticEdit
A catacaustic is the reflective case.
With a radiant, it is the evolute of the orthotomic of the radiant.
The planar, parallel-source-rays case: suppose the direction vector is <math>(a,b)</math> and the mirror curve is parametrised as <math>(u(t),v(t))</math>. The normal vector at a point is <math>(-v'(t),u'(t))</math>; the reflection of the direction vector is (normal needs special normalization)
- <math>2\mbox{proj}_nd-d=\frac{2n}{\sqrt{n\cdot n}}\frac{n\cdot d}{\sqrt{n\cdot n}}-d=2n\frac{n\cdot d}{n\cdot n}-d=\frac{
(av'^2-2bu'v'-au'^2,bu'^2-2au'v'-bv'^2) }{v'^2+u'^2}</math> Having components of found reflected vector treat it as a tangent
- <math>(x-u)(bu'^2-2au'v'-bv'^2)=(y-v)(av'^2-2bu'v'-au'^2).</math>
Using the simplest envelope form
- <math>F(x,y,t)=(x-u)(bu'^2-2au'v'-bv'^2)-(y-v)(av'^2-2bu'v'-au'^2)</math>
- <math>=x(bu'^2-2au'v'-bv'^2)
-y(av'^2-2bu'v'-au'^2) +b(uv'^2-uu'^2-2vu'v') +a(-vu'^2+vv'^2+2uu'v')</math>
- <math>F_t(x,y,t)=2x(bu'u-a(u'v+uv')-bv'v)
-2y(av'v-b(uv'+u'v)-au'u)</math>
- <math>+b( u'v'^2 +2uv'v -u'^3 -2uu'u -2u'v'^2 -2uvv' -2u'vv)
+a(-v'u'^2 -2vu'u +v'^3 +2vv'v +2v'u'^2 +2vuu' +2v'uu)</math> which may be unaesthetic, but <math>F=F_t=0</math> gives a linear system in <math>(x,y)</math> and so it is elementary to obtain a parametrisation of the catacaustic. Cramer's rule would serve.
ExampleEdit
Let the direction vector be (0,1) and the mirror be <math>(t,t^2).</math> Then
- <math>u'=1</math> <math>u=0</math> <math>v'=2t</math> <math>v=2</math> <math>a=0</math> <math>b=1</math>
- <math>F(x,y,t)=(x-t)(1-4t^2)+4t(y-t^2)=x(1-4t^2)+4ty-t</math>
- <math>F_t(x,y,t)=-8tx+4y-1</math>
and <math>F=F_t=0</math> has solution <math>(0,1/4)</math>; i.e., light entering a parabolic mirror parallel to its axis is reflected through the focus.
See alsoEdit
ReferencesEdit
External linksEdit
- {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web
|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Caustic%7CCaustic.html}} |title = Caustic |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}