---
tag: GMRYG4S3G0
kind: concept
name: Caustic
msc: [53D12, 58K05, 78A05]
sources:
  - url: https://en.wikipedia.org/wiki/Caustic_(mathematics)
same_as:
  wikidata: Q3492725
  wikipedia: Caustic (mathematics)
---

## Summary

A caustic is the [[envelope|827KR3MVVY]] of a family of rays, the curve or surface where nearby rays focus.
In symplectic terms, project a [[Lagrangian submanifold|H8ACPTMX08]] of a cotangent bundle to the base.
The caustic is the set of critical values of this projection, that is, the image of the [[Lagrangian singularities|QKPMR9MC9W]].
Caustics appear in optics as the bright curves of focused light.
They also appear in the asymptotics of oscillatory integrals and of short-wave solutions of wave equations.
Their generic local shapes, such as folds and cusps, are given by Arnold's classification of Lagrangian singularities.

## Explanations

### L1 human

On a sunny day the inside of a coffee cup shows a bright curve with a sharp point.
Light rays reflected from the curved wall crowd together along this curve, so it is much brighter than its surroundings.
That bright curve is a caustic: the envelope that the reflected rays touch.
A simple case from geometry is the family of normal lines of a plane curve, whose caustic is the [[evolute|Y3ZNV24V2C]] of the curve.
For a parabola the evolute has a cusp, and such cusps are the typical sharp points of caustics.