Caustic (mathematics) explained

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 where is a Lagrangian immersion of a Lagrangian submanifold L into a symplectic manifold M, and 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

(a,b) and the mirror curve is parametrised as (u(t),v(t)) . The normal vector at a point is ( - v'(t),u'(t)) ; the reflection of the direction vector is (normal needs special normalization)2 proj _{nd - d = (2 n)/(\sqrt{n ⋅ n)} }\frac-d=2n\frac-d=\fracHaving components of found reflected vector treat it as a tangent(x - u)(bu' ^{2-2au'v'-bv'} ^{2)=(y-v)(av'} ^{2-2bu'v'-au'} ^{2).} Using the simplest envelope formF(x,y,t) = (x - u)(bu' ^{2-2au'v'-bv'} ^{2)-(y-v)(av'} ^{2-2bu'v'-au'} ^{2)} = 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')} 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'')} + b( u'v' ^{2} +2 uv'v'' - u' ^{3} -2 uu'u'' -2 u'v' ^{2} -2 u''vv' -2 u'vv'') + a( - v'u' ^{2} -2 vu'u'' + v' ^{3} +2 vv'v'' +2 v'u' ^{2} +2 v''uu' +2 v'uu'') which may be unaesthetic, but F = F _{t=0} gives a linear system in (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

(t,t ^{2).} Thenu' =1   u'' =0   v' =2 t   v'' =2   a =0   b =1 F(x,y,t) = (x - t)( 1-4 t ^{2)+4t(y-t} ^{2)=x(1-4t} ^{2)+4ty-t} F _{t(x,y,t)=-8tx+4y-1} and F = F _{t=0} has solution ( 0 , 1 / 4 ) ; 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

Notes and References

1. Book: Arnold, V. I.. A. N.. Varchenko. S. M.. Gusein-Zade. Vladimir Arnold. Sabir Gusein-Zade. Alexander Varchenko. The Classification of Critical Points, Caustics and Wave Fronts: Singularities of Differentiable Maps, Vol 1. Birkhäuser. 1985. 0-8176-3187-9.
2. http://mathworld.wolfram.com/CircleCatacaustic.html Circle Catacaustic
3. Web site: Focusing on Nephroids. Levi. Mark. 2018-04-02. SIAM News. 2018-06-01.
4. http://atoptics.co.uk/fz552.htm Rainbow caustics
5. http://atoptics.co.uk/fz564.htm Caustic fringes