1.      Arfken, G., Mathematical Methods for Physicists (Academic Press, New York, 1970).


2.   Bentz, D. P. and Stutzman, P. E., "SEM Analysis and Computer Modelling of Hydration of Portland Cement Particles" Petrography of Cementitious Materials, in ASTM STP 1215, Sharon M. DeHayes and David Stark, Eds., American Society for Testing and Materials, Philadelphia, pp. 60-73, 1994.


3.      Bodziony, A., Gorsky, J., and Kraj W. (1975) “Determination of the surface area of the convex solid bodies by means of measuring the surface area of their shadows” Archimjn Cornictwa Tom XX Zessyt 4 pp 395-410.


4.      Bodziony, A.,  Gorsky, J., and Kraj, W. (1976) “On the method of determination of the surface area of Convex Bodies” Bulletin de L’Acamie Plonaise des Sciences v XXIV no.3.


5.      Cauchy, A. (1850) “Memoire sur la rectification des courbes et de la quadrature des surfaces courbes” Mem. Acad. Sci. Paris 22, no. 3.   Also in Oevres Completes Vol 1 (1908).


6.      Erdogan, S.T., Quiroga, P.N., Fowler, D.W., Saleh, H.A., Livingston, R.A., Garboczi, E.J., Ketcham, P.M., Hagedorn, J.G., and Satterfield, S.G., Three-dimensional shape analysis of coarse aggregates: Methodology and preliminary results on several different coarse aggregates, submitted to Cem. Conc. Res. (2004).


7.      Garboczi, E.J. (2002).  Three-dimensional mathematical analysis of particle shape using x-ray tomography and spherical harmonics: application to aggregates used in concreteCem. Conc. Res. 32, 1621-1638.


8.      Cheok, G.S., Stone, W.C., and Garboczi, E.J. (2005), “Using LADAR to characterize the 3-D shape of aggregates: Preliminary results,” submitted to Cem. Conc. Res.


9.      Garboczi, E.J., Douglas, J., and Bohn, R., “A hybrid finite element-analytical method for determining the intrinsic elastic moduli of particles having moderately extended shapes and a wide range of elastic properties,” Mech. of Materials (2005a), in press.


10.  Garboczi, E.J., and Douglas, J. (2005b), in preparation.


11.  Goldstein, H. (1950), Classical Mechanics (Addison-Wesley, Reading, MA, 1950).


12.  Jia, X. and Williams, R.A. (2001), “A packing algorithm for particles of arbitrary shape,” Powder Technol. 120, 175-186.


13.    Kak, A.C. and Slaney, M., Principles of Computerized Tomographic Imaging (SIAM, New York, 2001).


14.  Lau, T. (2002). “Using 2-d projections to characterize 3-d Particles” Thesis presented to the Faculty of the University of California, Davis in partial fulfillment of the requirements for the degree Master of Science in Civil Engineering June 2002.


15. Lawden, D.F. (1989) Elliptic Functions and Applications (Springer-Verlag, Berlin, 1989).


16. Legendre, A.-M. (1825) Traite des Fonctions Élliptiques, tome 1 (Huzard-Courchier, Paris, 1825).


17. Lin, C.L. and Miller, J.D. (2005), “3D characterization and analysis of particle shape using X-ray microtomography (XMT),” Powder Technol. 154, 61-69.


18. Maas, L.R.M. (1994), “On the surface area of an ellipsoid and related integrals of elliptic integrals,” J. Comp. Appl. Math. 51, 237-249. Note that Maas’ formula for the ellipsoid surface area has an incorrect prefactor for the term involving the elliptic E function of the second kind. Lawden’s reference has the correct prefactor.


19. Mandelbrot, B.B. (1967) The Fractal Geometry of Nature (W.H.Freeman, San Francisco, 1967).


20.  Mansfield, M.L., Covell, D.G., and Jernigan, R.L. (2002), “A new class of molecular shape descriptors. 1. Theory and properties,” J. Chem. Infor. Comp. Sci. 42, 259-273.


21. Mather, B. (1966) “Shape, Surface Texture and Coatings”.  ASTM STP 169A Significance of Tests and Properties of Concrete and Concrete Making Materials (American Society for Testing and Materials (ASTM), Philadelphia 1966), p. 571.


22. Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T., Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, Cambridge, 1989).


23. Russ J.C. (1999). The Image Processing Handbook, 3rd Ed. (CRC Pres, Boca Raton, Florida, 1999), p. 771.


24. Taylor, M. (2002) “Quantitative Measures of Shape and Size of Particles", Powder Technology 124 94-100.


25. Taylor, M. et al. (2005) “Using Projected Areas to Characterize 3-d irregular particles” Granite Rock Company, Watsonville, California. Report No MR 2005:1.


26. Thomsen, K. 2004 at See also Klamkin, M.S. "Elementary approximations to the area of n-dimensional ellipsoids", Amer. Math. Mon. 78 (1971) pp.280-283; “Corrections to Elementary approximations to the area of n-dimensional ellipsoids", ibid., 83 (1976) p. 478.


27. Umhauer, H. and Gutsch, A. (1997)  “Particle Characterization by Projected Area Determination,” Particles and Particle Systems Characterization 14, 105-115.


28. Underwood, E.E. (1970) Quantitative Stereology (Addison Wesley, New York, 1970).


29. Vickers, G.T. (1998) Powder Technol. 98, 250-257.



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