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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]}

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.

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 and the mirror curve is parametrised as . The normal vector at a point is ; the reflection of the direction vector is (normal needs special normalization)

Having components of found reflected vector treat it as a tangent

Using the simplest envelope form

which may be unaesthetic, but gives a linear system in and so it is elementary to obtain a parametrisation of the catacaustic. Cramer's rule would serve.

Let the direction vector be (0,1) and the mirror be Then

and has solution ; *i.e.*, light entering a parabolic mirror parallel to its axis is reflected through the focus.

**^**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".*SIAM News*. Retrieved 2018-06-01.**^**Rainbow caustics**^**Caustic fringes

- Weisstein, Eric W. "Caustic".
*MathWorld*.