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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...
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
Anomalies of the tides
, 2003
"... Unusual features of water waves that Galileo described in a letter to Cardinal Orsini in 1616 are revisited from the perspectives of singular optics and geometric analysis. ..."
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Unusual features of water waves that Galileo described in a letter to Cardinal Orsini in 1616 are revisited from the perspectives of singular optics and geometric analysis.
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.
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
AN ASYMPTOTIC UNIVERSAL FOCAL DECOMPOSITION FOR NONISOCHRONOUS POTENTIALS
, 908
"... Abstract. Galileo, in the XVII century, observed that the small oscillations of a pendulum seem to have constant period. In fact, the Taylor expansion of the period map of the pendulum is constant up to second order in the initial angular velocity around the stable equilibrium. It is well known that ..."
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Abstract. Galileo, in the XVII century, observed that the small oscillations of a pendulum seem to have constant period. In fact, the Taylor expansion of the period map of the pendulum is constant up to second order in the initial angular velocity around the stable equilibrium. It is well known that, for small oscillations of the pendulum and small intervals of time, the dynamics of the pendulum can be approximated by the dynamics of the harmonic oscillator. We study the dynamics of a family of mechanical systems that includes the pendulum at small neighbourhoods of the equilibrium but after long intervals of time so that the second order term of the period map can no longer be neglected. We analyze such dynamical behaviour through a renormalization scheme acting on the dynamics of this family of mechanical systems. The main theorem states that the asymptotic limit of this renormalization scheme is universal: it is the same for all the elements in the considered class of mechanical systems. As a consequence, we obtain an universal asymptotic focal decomposition for this family of mechanical systems. This paper is intended to be the first of a series of articles aiming at a semiclassical quantization of systems of the pendulum type as a natural application of the focal decomposition associated to the twopoint boundary value problem. 1.
CWP500P Spurious multiples in seismic interferometry of primaries
"... Seismic interferometry yields the Green’s function that accounts for wave propagation between receivers by correlating the waves recorded at these receivers. We present a derivation of this principle based on the method of stationary phase. Although this derivation is applicable to simple media only ..."
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Seismic interferometry yields the Green’s function that accounts for wave propagation between receivers by correlating the waves recorded at these receivers. We present a derivation of this principle based on the method of stationary phase. Although this derivation is applicable to simple media only, it provides insight into the physical principle of seismic interferometry. In a homogeneous medium with one horizontal reflector and without a free surface, the correlation of the waves recorded at two receivers correctly gives both the direct wave and the singlereflected waves. When more reflectors are present a product of the singlereflected waves occurs in the cross correlation that leads to spurious multiples when the waves are excited at the surface only. We give a heuristic argument that these spurious multiples disappear when sources below the reflectors are included. We also extend the derivation to a smoothly varying inhomogeneous background medium. Key words: siesmic interferometry, stationary phase, multiples 1
The Rainbow Bridge: Rainbows in Art, Myth, and
"... This is a magnificent and scholarly book, exquisitely produced, and definitely not destined only for the coffee table. It is multifaceted in character, addressing rainbowrelevant aspects of mythology, religion, the history of art, art criticism, the history of optics, the theory of color, the philo ..."
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This is a magnificent and scholarly book, exquisitely produced, and definitely not destined only for the coffee table. It is multifaceted in character, addressing rainbowrelevant aspects of mythology, religion, the history of art, art criticism, the history of optics, the theory of color, the philosophy of science, and advertising! The quality of the reproductions and photographs is superb. The authors are experts in meteorological optics, but their book draws on many other subdisciplines. It is a challenge, therefore, to write a review about a book that contains no equations or explicit mathematical themes for what is primarily a mathematical audience. However, while the mathematical description of the rainbow may be hidden in this book, it is nonetheless present. Clearly, such a review runs the risk of giving a distorted picture of what the book is about, both by “unfolding ” the hidden mathematics and suppressing, to some extent, John A. Adam is professor of mathematics at Old Dominion