Abstract
This paper considers an ellipse, produced by the intersection of a triaxial ellipsoid and a plane (both arbitrarily oriented), and derives explicit expressions for its axis ratio and orientation. The relationship between central and noncentral sections, and the problem of intersection probability, are discussed. The resulting equations have obvious applications in structural geology and optical mineralogy, and in stereological modeling of grain-size and grain-shape distributions.
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References
Bell, R. J. T., 1912, Coordinate geometry of three dimensions, 2nd ed: Macmillan, London, 394 p.
DeHoff, R. T., 1962, The determination of the size distribution of ellipsoidal particles from measurements made on random plane sections: Trans. AIME, v. 224, p. 474–477.
DeHoff, R. T. and Bousquet, P., 1970, Estimation of the size distribution of triaxial ellipsoidal particles from the distribution of linear intercepts: Jour. Microscopy, v. 92, p. 119–135.
DeHoff, R. T. and Rhines, F. N., 1961, Determination of number of particles per unit volume from measurements made on random plane sections: the general cylinder and the ellipsoid: Trans. AIME, v. 221, p. 975–982.
Flinn, D., 1978, Construction and computation of three-dimensional progressive deformations: Jour. Geol. Soc. Lond., v. 135, p. 291–305.
Hennig, A. and Elias, H., 1971, A rapid method for the visual determination of size distribution of spheres from the size distribution of their sections: Jour. Microscopy, v. 93, p. 101–107.
Jones, K. A. and Galwey, A. K., 1966, Size distribution, composition, and growth kinetics of garnet crystals in some metamorphic rocks from the West of Ireland: Q. Jour. Geol. Soc. Lond., v. 122, p. 29–44.
Kellerhals, R., Shaw, J., and Arora, V. K., 1975, On grain size from thin sections: J. Geol., v. 83, p. 79–96.
Kretz, R., 1966, Grain-size distribution for certain metamorphic minerals in relation to nucleation and growth: J. Geol., v. 74, p. 147–173.
Ramsay, J. G., 1967, Folding and fracturing of rocks: McGraw-Hill, New York, 568 p.
Salmon, G., 1914, Analytic geometry of three dimensions, 6th ed: Chelsea, New York, 470 p.
Wicksell, S. D., 1925, The corpuscle problem: second memoir; case of ellipsoidal corpuscles: Biometrika, v. 17, p. 152–170.
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Ferguson, C.C. Intersections of ellipsoids and planes of arbitrary orientation and position. Mathematical Geology 11, 329–336 (1979). https://doi.org/10.1007/BF01034997
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DOI: https://doi.org/10.1007/BF01034997