Minimum Path Problems in Normed Spaces, Reflection and Refraction

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Minimum Path Problems in Normed Spaces, Reflection and Refraction

Show simple item record Golomb, Michael en Ghandehari, Mostafa en 2010-06-08T17:38:48Z en 2010-06-08T17:38:48Z en 1996 en
dc.identifier.uri en
dc.description.abstract The main minimum (or extremum) path problem in this paper deals with the "law of refraction" at a curve separating the plane into two parts with different norms. Analytic and geometric characterization for the point at which refraction takes place and formulas for the angles that this incident and refracted rays make with a fixed axis or with the normal to the curve are established. The case where the unit circle of the two norms are Euclidean circles with different radii leads to the traditional Snell's Law. The other problem deals with the "law of reflection" from a curve in the normed plane, which in the case of Euclidean norm asserts the equality of the angles of incidence and reflection. The 3-dimensional case, where the separating curve is replaced by a surface, is also considered. Finally it is shown that minimization of path length with respect to non-Euclidean norms is not a special case of Fermat's principle of minimizing the line integral [see pdf for notation] for a suitable refraction index n. en
dc.language.iso en_US en
dc.publisher University of Texas at Arlington en
dc.relation.ispartofseries Technical Report;309 en
dc.subject Fermat's principle en
dc.subject Snell's law en
dc.subject minimization of path length en
dc.subject Normed linear space en
dc.subject.lcsh Electromagnetic waves en
dc.subject.lcsh Mathematics Research en
dc.title Minimum Path Problems in Normed Spaces, Reflection and Refraction en
dc.type Technical Report en
dc.publisher.department Department of Mathematics en

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