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.
A catacaustic is the reflective case.
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
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.