# Caustic (mathematics)

__: Envelope of rays either reflected or refracted by a manifold__

**Short description**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 (*π* ○ *i*) : *L* ↪ *M* ↠ *B*; where *i* : *L* ↪ *M* is a Lagrangian immersion of a Lagrangian submanifold *L* into a symplectic manifold *M*, and *π* : *M* ↠ *B* is a Lagrangian fibration of the symplectic manifold *M*. The caustic is a subset of the Lagrangian fibration's base space *B*.^{[1]}

## Explanation

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.^{[2]}^{[3]} Rippling caustics are commonly formed when light shines through waves on a body of water.

Another familiar caustic is the rainbow.^{[4]}^{[5]} Scattering of light by raindrops causes different wavelengths of light to be refracted into arcs of differing radius, producing the bow.

## Catacaustic

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]\displaystyle{ (a,b) }[/math] and the mirror curve is parametrised as [math]\displaystyle{ (u(t),v(t)) }[/math]. The normal vector at a point is [math]\displaystyle{ (-v'(t),u'(t)) }[/math]; the reflection of the direction vector is (normal needs special normalization)

- [math]\displaystyle{ 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]\displaystyle{ (x-u)(bu'^2-2au'v'-bv'^2)=(y-v)(av'^2-2bu'v'-au'^2). }[/math]

Using the simplest envelope form

- [math]\displaystyle{ F(x,y,t)=(x-u)(bu'^2-2au'v'-bv'^2)-(y-v)(av'^2-2bu'v'-au'^2) }[/math]
- [math]\displaystyle{ =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]\displaystyle{ F_t(x,y,t)=2x(bu'u''-a(u'v''+u''v')-bv'v'')
-2y(av'v''-b(u''v'+u'v'')-au'u'') }[/math]
- [math]\displaystyle{ +b( u'v'^2 +2uv'v'' -u'^3 -2uu'u'' -2u'v'^2 -2u''vv' -2u'vv'') +a(-v'u'^2 -2vu'u'' +v'^3 +2vv'v'' +2v'u'^2 +2v''uu' +2v'uu'') }[/math]

which may be unaesthetic, but [math]\displaystyle{ F=F_t=0 }[/math] gives a linear system in [math]\displaystyle{ (x,y) }[/math] and so it is elementary to obtain a parametrisation of the catacaustic. Cramer's rule would serve.

### Example

Let the direction vector be (0,1) and the mirror be [math]\displaystyle{ (t,t^2). }[/math] Then

- [math]\displaystyle{ u'=1 }[/math] [math]\displaystyle{ u''=0 }[/math] [math]\displaystyle{ v'=2t }[/math] [math]\displaystyle{ v''=2 }[/math] [math]\displaystyle{ a=0 }[/math] [math]\displaystyle{ b=1 }[/math]
- [math]\displaystyle{ F(x,y,t)=(x-t)(1-4t^2)+4t(y-t^2)=x(1-4t^2)+4ty-t }[/math]
- [math]\displaystyle{ F_t(x,y,t)=-8tx+4y-1 }[/math]

and [math]\displaystyle{ F=F_t=0 }[/math] has solution [math]\displaystyle{ (0,1/4) }[/math]; *i.e.*, light entering a parabolic mirror parallel to its axis is reflected through the focus.

## See also

- Cut locus (Riemannian manifold)
- Last geometric statement of Jacobi
- Nephroid caustic

## References

- ↑ Arnold, V. I.; Varchenko, A. N.; Gusein-Zade, S. M. (1985).
*The Classification of Critical Points, Caustics and Wave Fronts: Singularities of Differentiable Maps, Vol 1*. Birkhäuser. ISBN 0-8176-3187-9. - ↑ Circle Catacaustic. Wolfram MathWorld. Retrieved 2009-07-17.
- ↑ Levi, Mark (2018-04-02). "Focusing on Nephroids". https://sinews.siam.org/Details-Page/focusing-on-nephroids.
- ↑ Rainbow caustics
- ↑ Caustic fringes

## External links

Original source: https://en.wikipedia.org/wiki/Caustic (mathematics).
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