Minimum Path Problems in Normed Spaces: Reflection and Refraction

M. Ghandehari and M Golomb
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M. Ghandehari: University of Texas at Arlington
M Golomb: Purdue University

Journal of Optimization Theory and Applications, 2000, vol. 105, issue 1, No 1, 16 pages

Abstract: Abstract The paper deals with the existence and characterization of minimum or extremum paths connecting two given points in a vector space, which is divided by a barrier (a curve C if the space is 2-dimensional) into two parts with different norms. The global problem of existence of polygonal paths of shortest length is dealt with in Section 2. An example shows that, for a curve with a point of inflection, such paths may not exist. However, the existence of such paths is proved for a more restricted class of curves (Theorem 2.3). The notion of permissible polygonal paths is introduced, and it is shown that, for a very general class of curves, such paths of shortest length do exist (Theorem 2.2). Sections 3 and 4 deal with the local conditions at the intersection of the extremal path with the curve C. Theorem 4.1 establishes a geometric characterization of the point of intersection, and Eqs. (13) and (15) are formulas for the angles that the segments of the extremal path make with a fixed axis or with the normal to C at the point of intersection. The case where the unit circles of the tax norms are Euclidean circles with different radii leads to the traditional Snell law. Section 6 deals with the law of reflection at the curve C, which in the case of the Euclidean norm asserts the equality of the angles of incidence and reflection. The n-dimensional case, where the curve C is replaced by a hypersurface, is considered briefly in Section 7.

Keywords: normed linear spaces; barriers; minimization of path lengths; polygonal paths; permissible polygonal paths; Snell law; Fermat principle (search for similar items in EconPapers)
Date: 2000
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DOI: 10.1023/A:1004684626225

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