In differential geometry a caustic is the envelope of rays either reflected or refracted by a manifold. Obviously it is related to the optical concept of caustics. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... The form of envelope treated here is a manifold that manages to be tangent to some point of each member of a family of manifolds. ... Ray may refer to: An electrical ray: see laser. ... IT IS KNOWN AS MARK a lunitice insain int gw brain ... This article refers to refraction in waves. ... This page is about a higher mathematics topic. ... The word caustics has several meanings depending upon the context in which it is used: In Greek language, from which this word originates, caustics means to burn or burning. In chemistry a caustic substance is one that eats away or chemically burns other materials by process of attacking it basically...

The ray's source may be a point (called the radiant) or infinity, in which case a direction vector must be specified.

Catacaustic

A catacaustic is the reflective case.

With a radiant, it is the evolute of the orthotomic of the radiant. In the differential geometry of curves, the evolute of a curve is the set of all its centers of curvature. ... The orthotomic is the set of reflections of a given point around the tangents of a given curve. ...

The planar, parallel-source-rays case: suppose the direction vector is (a,b) and the mirror curve is parametrised as (u(t),v(t)). The normal vector at a point is ( − v'(t),u'(t)); the reflection of the direction vector is

so the reflected ray satisfies

(x − u)(bu'² − 2au'v' − bv'²) = (y − v)(av'² − 2bu'v' − au'²).

Using the simplest envelope form The form of envelope treated here is a manifold that manages to be tangent to some point of each member of a family of manifolds. ...

F(x,y,t) = (x − u)(bu'² − 2au'v' − bv'²) − (y − v)(av'² − 2bu'v' − au'²) = x(bu'² − 2au'v' − bv'²) − y(av'² − 2bu'v' − au'²) + b(uv'² − uu'² − 2vu'v') + a( − vu'² + vv'² + 2uu'v')

F_t(x,y,t) = 2x(bu'u'' − a(u'v'' + u''v') − bv'v'') − 2y(av'v'' − b(u''v' + u'v'') − au'u'') + b(u'v'² + 2uv'v'' − u'³ − 2uu'u'' − 2u'v'² − 2u''vv' − 2u'vv'') + a( − v'u'² − 2vu'u'' + v'³ + 2vv'v'' + 2v'u'² + 2v''uu' + 2v'uu'')

which looks horrid, but F = F_t = 0 gives a linear system in (x,y) and so it is elementary to obtain a parametrisation of the catacaustic. Cramer's rule would serve. A linear system is a model of a system based on some kind of linear operator. ... Cramers rule is a theorem in linear algebra, which gives the solution of a system of linear equations in terms of determinants. ...

Example

Let the direction vector be (0,1) and the mirror be (t,t²). Then

u' = 1   u'' = 0   v' = 2t   v'' = 2   a = 0   b = 1

F(x,y,t) = (x − t)(1 − 4t²) + 4t(y − t²) = x(1 − 4t²) + 4ty − t

F_t(x,y,t) = − 8tx + 4y − 1

and F = F_t = 0 has solution (0,1 / 4); i.e., light entering a parabolic mirror parallel to its axis is reflected through the focus. A parabola The parabola (from the Greek: παραβολή) is a conic section generated by the intersection of a cone and a plane tangent to the cone or parallel to some plane tangent to the cone. ...

Diacaustic

A diacaustic is the refractive case. It is complicated by the need for another datum (refractive index) and the fact that refraction is not linear -- Snell's law is "ugly" in pure vector notation. The word linear comes from the Latin word linearis, which means created by lines. ... Snells law is the simple formula used to calculate the refraction of light when travelling between two media of differing refractive index. ...

External links

• Mathworld