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Illumination from Curved Reflectors
, 1992
"... A technique is presented to compute the reflected illumination from curved mirror surfaces onto other surfaces. In accordance with Fermat's principle, this is equivalent to finding extremal paths from the light source to the visible surface via the mirrors. Once pathways of illumination are fou ..."
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A technique is presented to compute the reflected illumination from curved mirror surfaces onto other surfaces. In accordance with Fermat's principle, this is equivalent to finding extremal paths from the light source to the visible surface via the mirrors. Once pathways of illumination are found, irradiance is computed from the Gaussian curvature of the geometrical wavefront. Techniques from optics, differential geometry and interval analysis are applied to solve these problems. CR Categories and Subject Descriptions: I.3.3 [ Computer Graphics ]: Picture/Image Generation; I.3.7 [ Computer Graphics ]: ThreeDimensional Graphics and Realism General Terms: Algorithms Additional Keywords and Phrases: Caustics, Differential Geometry, Geometrical Optics, Global Illumination, Interval Arithmetic, Ray Tracing, Wavefronts 1. Introduction Ray tracing provides a straightforward means for synthesizing realistic images on the computer. A scene is first modeled, usually by a collection of implici...
Gravitational Lensing from a Spacetime Perspective
 Living Rev. Relativity
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The optical singularities of birefringent dichroic chiral crystals
 A
"... Using a new formalism involving projection from the sphere of directions to the stereographic plane, and associated complex variables, explicit formulae are obtained for the two refractive indices and polarizations in optically anisotropic crystals that are both dichroic (absorbing) and chiral (opti ..."
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Using a new formalism involving projection from the sphere of directions to the stereographic plane, and associated complex variables, explicit formulae are obtained for the two refractive indices and polarizations in optically anisotropic crystals that are both dichroic (absorbing) and chiral (optically active). This enables three types of polarization singularity to be classified and explored: singular axes, which are degeneracies where the two refractive indices are equal, and which for a transparent nonchiral crystal condense pairwise onto the optic axes; C points, where the polarization is purely circular (right or lefthanded), with topological index +1, +12 or + 1 4 and whose positions are independent of the chirality; and L lines, where the polarization is purely linear, dividing direction space into regions with rightand lefthandedness. A local model captures essential features of the general theory. Interference figures generated by slabs of crystal viewed directly or through a polarizer and/or analyser enable the singularities to be displayed directly.
The theory of caustics and wavefront singularities with physical applications
 Jounral of Mathematical Physics A
"... This is intended as an introduction to and review of the work of V. I. Arnold and his collaborators on the theory of Lagrangian and Legendrian submanifolds and their associated maps. The theory is illustrated by applications to HamiltonJacobi theory and the eikonal equation, with an emphasis on nul ..."
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This is intended as an introduction to and review of the work of V. I. Arnold and his collaborators on the theory of Lagrangian and Legendrian submanifolds and their associated maps. The theory is illustrated by applications to HamiltonJacobi theory and the eikonal equation, with an emphasis on null surfaces and wavefronts and their associated caustics and singularities. 1 I.
Caustics, multiply reconstructed by Talbot interference
, 1999
"... In planar geometrical optics, the rays normal to a periodically undulating wavefront curve W generate caustic lines that begin with cusps and recede to infinity in pairs; therefore these caustics are not periodic in the propagation distance z. On the other hand, in paraxial wave optics the phase dif ..."
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In planar geometrical optics, the rays normal to a periodically undulating wavefront curve W generate caustic lines that begin with cusps and recede to infinity in pairs; therefore these caustics are not periodic in the propagation distance z. On the other hand, in paraxial wave optics the phase diffraction grating corresponding to W gives a pattern that is periodic in z, the period for wavelength / and grating period a being the Talbot distance, ZT = a2//, that becomes infinite in the geometrical limit. A model where W is sinusoidal gives a oneparameter family of diffraction fields, which we explore with numerical simulations, and analytically, to see how this clash of limits (that wave optics is periodic but ray optics is not) is resolved. The geometrical cusps are reconstructed by interference, not only at integer multiples of ZT but also, according to the fractional Talbot effect, at rational multiples of z = ZTp/q, in groups of q cusps within each grating period, down to a resolution scale set by/. In addition to caustics, the patterns show dark lanes, explained in detail by an averaging argument involving interference.
Making light of mathematics
 Bull. A.M.S.,40
"... According to the J. W. Gibbs Fan Club Homepage [51], Gibbs “was a quiet, dignified man whose life was presumably boring (part of the reason why no one has heard of him)”, and “a physicist, chemist, and mathematician—a sort of allaround hard sciences guy. ” Gibbs was a mathematical physicist, so at ..."
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According to the J. W. Gibbs Fan Club Homepage [51], Gibbs “was a quiet, dignified man whose life was presumably boring (part of the reason why no one has heard of him)”, and “a physicist, chemist, and mathematician—a sort of allaround hard sciences guy. ” Gibbs was a mathematical physicist, so at least the second
Internal caustic structure of illuminated liquid droplets
, 1990
"... and is made available as an electronic reprint with the permission of OSA. The paper can be found at the following ..."
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and is made available as an electronic reprint with the permission of OSA. The paper can be found at the following
Symmetric array of offaxis singular beams: spiral beams and their critical points
, 2007
"... We consider conditions of structural stability under which the array of singular beams preserves its topological structure and intensity distribution while slightly perturbing its intrinsic parameters. The orbital angular momentum of the array as a function of the array parameters is a characteristi ..."
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We consider conditions of structural stability under which the array of singular beams preserves its topological structure and intensity distribution while slightly perturbing its intrinsic parameters. The orbital angular momentum of the array as a function of the array parameters is a characteristic function, and its extreme points correspond to stable and unstable array states. © 2007 Optical Society of America OCIS codes: 140.3300, 260.3160, 350.5030. 1.
On the Specialness of Special Functions (The Nonrandom Effusions of the Divine Mathematician)
, 2007
"... This article attempts to address the problem of the applicability of mathematics in physics by considering the (narrower) question of what make the socalled special functions of mathematical physics special. It surveys a number of answers to this question and argues that neither simple pragmatic an ..."
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This article attempts to address the problem of the applicability of mathematics in physics by considering the (narrower) question of what make the socalled special functions of mathematical physics special. It surveys a number of answers to this question and argues that neither simple pragmatic answers, nor purely mathematical classificatory schemes are sufficient. What is required is some connection between the world and the way investigators are forced to represent the world.
Disruption of images: the caustictouching theorem 1. INTROLiUCTION
, 1986
"... Because of the distortions of geometrical optics,.image curves< and, the outlines of the objects that generate them need not have the same topology. New image loops appear when the object curve touches the caustic of the family of (imaginary) rays emitted by the observing eye. Such disruption may ..."
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Because of the distortions of geometrical optics,.image curves< and, the outlines of the objects that generate them need not have the same topology. New image loops appear when the object curve touches the caustic of the family of (imaginary) rays emitted by the observing eye. Such disruption may be elliptic (loop born from an isolated point) or hyperbolic (loop pinched off from an already existing one). The number of images need not be odd (unlike the number of rays reaching the eye from each object point)..Two examples are employed to illustrate caustic touching. The first is the Sun’s disk seen in rippled water (as the height of the eye increases, the boundary of all the images becomes a fractal curve with dimension 2). The second is sunset seen through the Earth’s atmosphere from near space (when there is an inversion layer) or from the Moon during a lunar eclipse (when there need not be one). Images formed by light rays are nearly always distorted representations of their objects. Optical instruments ark, of course, designed to minimize the geonietrical aberrations respdnsible for distortion, but with nature’s optics, whose elements may be the reflecting surface of rippled water, or refractiveindex gradients in the atmosphere, severe distortions are unavoidable. An extreme of distortion occurs