Caustic (mathematics)
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In differential geometry, a caustic is the
envelope An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card. Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a ...
of rays either
reflected Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...
or
refracted In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenomeno ...
by a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
. It is related to the concept of caustics in
geometric optics Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
. 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 Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...
and
singularity theory In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...
, a caustic is the critical value set of a Lagrangian mapping where is a Lagrangian immersion of a
Lagrangian submanifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sympl ...
''L'' into a
symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called s ...
''M'', and is a Lagrangian fibration of the symplectic manifold ''M''. The caustic is a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of the Lagrangian
fibration The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or obstruction theory. In this article, all map ...
's base space ''B''.


Explanation

Concentration of light, especially sunlight, can burn. The word ''caustic'', in fact, comes from the Greek καυστός, burnt, via the Latin ''causticus'', burning. A common situation where caustics are visible is when light shines on a drinking glass. The glass casts a shadow, but also produces a curved region of bright light. In ideal circumstances (including perfectly parallel rays, as if from a point source at infinity), a
nephroid In geometry, a nephroid () is a specific plane curve. It is a type of epicycloid in which the smaller circle's radius differs from the larger by a factor of one-half. Name Although the term ''nephroid'' was used to describe other curves, it was ...
-shaped patch of light can be produced. Rippling caustics are commonly formed when light shines through waves on a body of water. Another familiar caustic is the
rainbow A rainbow is a meteorological phenomenon that is caused by reflection, refraction and dispersion of light in water droplets resulting in a spectrum of light appearing in the sky. It takes the form of a multicoloured circular arc. Rainbows ...
.Caustic fringes
/ref> Scattering of light by raindrops causes different
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
s of light to be refracted into arcs of differing radius, producing the bow.


Catacaustic

A catacaustic is the reflective case. With a radiant, it is the
evolute In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that c ...
of the orthotomic of the radiant. 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 (normal needs special normalization) :2\mbox_nd-d=\frac\frac-d=2n\frac-d=\frac Having components of found reflected vector treat it as a tangent :(x-u)(bu'^2-2au'v'-bv'^2)=(y-v)(av'^2-2bu'v'-au'^2). Using the simplest
envelope An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card. Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a ...
form :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 may be unaesthetic, but F=F_t=0 gives a
linear system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstracti ...
in (x,y) and so it is elementary to obtain a parametrisation of the catacaustic.
Cramer's rule In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants ...
would serve.


Example

Let the direction vector be (0,1) and the mirror be (t,t^2). Then :u'=1   u''=0   v'=2t   v''=2   a=0   b=1 :F(x,y,t)=(x-t)(1-4t^2)+4t(y-t^2)=x(1-4t^2)+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.


See also

*
Cut locus (Riemannian manifold) In Riemannian geometry, the cut locus of a point p in a manifold is roughly the set of all other points for which there are multiple minimizing geodesics connecting them from p, but it may contain additional points where the minimizing geodesic is u ...
* Last geometric statement of Jacobi * Nephroid caustic


References


External links

* {{Differential transforms of plane curves Differential geometry