1. Catacaustic
    Example 

2. References

3. See also

4. External links

In differential geometry and geometric optics, a caustic is the envelope of rays either reflected or refracted by a manifold. It is related to the concept of caustics in 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 {{nowrap|1=(π ○ i) : L ↪ M ↠ B;}} where {{nowrap|1=i : L ↪ M}} is a Lagrangian immersion of a Lagrangian submanifold L into a symplectic manifold M, and {{nowrap|1=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]

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 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.

Example

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.

References

See also

• Cut locus (Riemannian manifold)
• Last geometric statement of Jacobi

External links

• {{MathWorld|title=Caustic|urlname=Caustic}}

• Mirazi Khel Dynasty
• Mir Aziz
• Mir Azizi
• Mir Azizi, Eslamabad-e Gharb
• Mir Azizi, Ravansar
• Mir Azizi, Sahneh
• Mir Azizi-ye Qadim
• Miraziz Jalilov
• Miraziz Jalolov
• Miraç Kal
• Miraña
• Miraña Indians
• Miraña language
• Miraş River
• Miraševac
• Mira Żelechower-Aleksiun
• Mirbacheh
• Mirbag-e Jonubi
• Mirbag-e Jonubi Rural District
• Mirbag-e Shomali
• Mirbag-e Shomali Rural District
• Mirbaha od Din
• Mirbaha ol Din
• Mir Bajgan
• Mir Bangesh