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Topology and geometry of caustics in relation with experiments

100%
Joets A.
Banach Center Publications
|
1999
|
tom 50
|
nr 1
169-177
EN
Caustics of geometrical optics are understood as special types of Lagrangian singularities. In the compact case, they have remarkable topological properties, expressed in particular by the Chekanov relation. We show how this relation may be experimentally checked on an example of biperiodic caustics produced by the deflection of the light by a nematic liquid crystal layer. Moreover the physical laws may impose a geometrical constraint, when the system is invariant by some group of symmetries. We show, on the example of polyhedral caustics, how the two constraints force degenerate umbilics of integer index to appear and determine their spatial organization.
2
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Caustics in Greek antiquity

100%
Joets A.
Banach Center Publications
|
2008
|
tom 82
|
nr 1
157-161
EN
The word caustic was introduced by Tschirnhausen in 1686, in the Latin expression caustica curva. We show that the study of the optical caustics goes back well before, at least to the hellenistic period. We present a small Greek text, whose author is perhaps Geminus (1st cent. B.C.), describing an optical phenomenon called achilles. We show that the term achilles, which has appeared only once, to our knowledge, in the literature, means caustics by reflection. We complete the description of the achilles thanks to another text, a passage of the poem Argonautika of Apollonius Rhodius. Finally, we attempt to explain the association between the mythical hero Achilles and the optical phenomenon called achilles.
3
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Singularities in drawings of singular surfaces

100%
Joets A.
Banach Center Publications
|
2008
|
tom 82
|
nr 1
143-156
EN
When drawing regular surfaces, one creates a concrete and visual example of a projection between two spaces of dimension 2. The singularities of the projection define the apparent contour of the surface. As a result there are two types of generic singularities: fold and cusp (Whitney singularities). The case of singular surfaces is much more complex. A priori, it is expected that new singularities may appear, resulting from the "interaction" between the singularities of the surface and the singularities of the projection. The problem has already been solved for the projection of a surface with a boundary. We consider here additional examples: the drawing of caustics and the drawing of the eversion of a sphere.
4
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Symmetric caustics and Curie's principle

51%
Joets A., Belaidi A., Ribotta R.
Banach Center Publications
|
2003
|
tom 62
|
nr 1
135-142
EN
Physical systems producing caustics may possess symmetries. In that case the relation between the symmetry of the system, considered as a whole, and the symmetry of the caustic follow a very general symmetry principle, the Curie principle. We give various examples of application of the Curie principle to caustics produced by the deflection of light in liquid crystals: the so called squint effect, the visualization of a new type of roll structure, etc. We show also that the Curie principle applies to physical systems having multiple stable states (variants).
5
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Singularities, defects and chaos in organized fluids

51%
Ribotta R., Belaidi A., Joets A.
Banach Center Publications
|
2003
|
tom 62
|
nr 1
223-238
EN
The singularities occurring in any sort of ordering are known in physics as defects. In an organized fluid defects may occur both at microscopic (molecular) and at macroscopic scales when hydrodynamic ordered structures are developed. Such a fluid system serves as a model for the study of the evolution towards a strong disorder (chaos) and it is found that the singularities play an important role in the nature of the chaos. Moreover both types of defects become coupled at the onset of turbulence. Besides this specificity, the results can be generalized to any structured physical system. They tend to demonstrate that the full knowledge of the system is "contained inside the surroundings of the singularity". It is also shown that such defects play a crucial role in all types of transitions between homogeneously ordered states from the rest state up to chaos.
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