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Caustic (mathematics)

## Summary

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 (πi) : LMB; where i : LM is a Lagrangian immersion of a Lagrangian submanifold L into a symplectic manifold M, and π : MB 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 ${\displaystyle (a,b)}$  and the mirror curve is parametrised as ${\displaystyle (u(t),v(t))}$ . The normal vector at a point is ${\displaystyle (-v'(t),u'(t))}$ ; the reflection of the direction vector is (normal needs special normalization)

${\displaystyle 2{\mbox{proj}}_{n}d-d={\frac {2n}{\sqrt {n\cdot n}}}{\frac {n\cdot d}{\sqrt {n\cdot n}}}-d=2n{\frac {n\cdot d}{n\cdot n}}-d={\frac {(av'^{2}-2bu'v'-au'^{2},bu'^{2}-2au'v'-bv'^{2})}{v'^{2}+u'^{2}}}}$

Having components of found reflected vector treat it as a tangent

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

Using the simplest envelope form

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

which may be unaesthetic, but ${\displaystyle F=F_{t}=0}$  gives a linear system in ${\displaystyle (x,y)}$  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 ${\displaystyle (t,t^{2}).}$  Then

${\displaystyle u'=1}$    ${\displaystyle u''=0}$    ${\displaystyle v'=2t}$    ${\displaystyle v''=2}$    ${\displaystyle a=0}$    ${\displaystyle b=1}$
${\displaystyle F(x,y,t)=(x-t)(1-4t^{2})+4t(y-t^{2})=x(1-4t^{2})+4ty-t}$
${\displaystyle F_{t}(x,y,t)=-8tx+4y-1}$

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