The common conjugate directions of plane sections of two concentric ellipsoids. (English) Zbl 0829.73026

Summary: We consider the problem of determining, for a given ellipsoid, the directions of the principal axes of any plane elliptical section (problem I), and present a generalization of this problem concerning a pair of concentric ellipsoids. It consists in determining the common conjugate directions of the pair of ellipses in which any plane cuts the two ellipsoids (problem II). After recalling the known results of problem I and presenting them in a form suitable to their generalization, we derive the solutions of problem II. Also, an appropriate transformation is introduced so that the results of problem II may be obtained from the corresponding results of problem I. These problems are considered here because they arise when dealing with plane wave propagation in crystal optics or crystal acoustics.


74J10 Bulk waves in solid mechanics
74E10 Anisotropy in solid mechanics
78A40 Waves and radiation in optics and electromagnetic theory