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CAUSTIC SURFACE

The term caustic refers, in a general fashion, to the envelope of the light rays emited by a point at finite distance (the source is then called the *radiant*) or at infinite distance after they are modified by an optical instrument. Every modified ray is considered as a whole, and includes the virtual ray.

The *caustic surface by reflection* (or *catacaustic surface*) of a surface with respect to a light source *S* is the envelope of the rays emitted by *S* after reflection on a , considered to be a mirror.

Example: caustic at infinite distance of a developable surface.

In this case, the incident rays along a generatrix form a plane called the incident plane of this generatrix; the reflected rays also form a plane which is the symmetric image of the incident plane with respect to the common normal plane along this generatrix; then, the caustic surface is the developable surface that envelopes the reflected planes.

In the case of a cylinder, the caustic surface is the cylinder the directrix of which is the caustic at infinite distance of the directrix of the initial cylinder, the incident rays being the sections of the incident planes by the plane of the directrix.

In the case of a cone with vertex *S* and directrix () in a plane *P*, the caustic surface is a cone with vertex *S* the directrix of which, in the plane *P*, is the caustic of the curve () with respect to the point *A*, intersection between the incident ray passing by *S* and *P*: note that it is therefore a caustic at finite distance.

In particular, when the cone is a surface of revolution, then the directrix is the caustic of a circle, hence the evolute of a limaçon of Pascal; it is therefore a cardioid when the incident rays are parallel to a generatrix of the cone.

When the rays are parallel to a generatrix of the container, the planar section of the caustic surface is a cardioid.

Be careful, the focal of a surface is also sometimes referred to as its caustic.

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© Robert FERRÉOL, Jacques MANDONNET 2017