# Ellipsoid Targeting with Overlap

## Abstract

First, we investigate the possibility of destruction of a passive point target. Subsequently, we study the problem of determination of best targeting points in an area within which stationary or mobile targets are distributed uniformly or normally. Partial results are given in the case in which the number of targeting points is less than seven or four, respectively. Thereafter, we study the case where there is no information on the enemy distribution. Then, the targeting should be organized in such a way that the surface defined by the kill radii of the missiles fully covers each point within a desired region of space-time. The problem is equivalent to the problem of packing ellipsoids of different sizes and shapes into an ellipsoidal container in \(\mathbb{R}^{4}\) so as to minimize a measure of overlap between ellipsoids is considered.

## Keywords

Function of damage Distribution of targets Ellipsoid targeting with overlap Sphere packing Ellipsoid packing Overlap measure## AMS subject classifications 2010:

65K10 90C22 65C50## References

- 1.M. Abramowitz, I.A. Stegun (eds.),
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*. Applied Mathematical Series, vol. 55 (National Bureau of Standards, Washington, DC, 1964)Google Scholar - 2.A.V. Bondareno, D.P. Hardin, E.B. Saff, Minimal N-point diameters and f-best-packing constants in \(\mathbb{R}^{d}\). Proc. AMS
**142**(3), 981–988 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 3.J. Borwein, A.S. Lewis,
*Convex Analysis and Nonlinear Optimization: Theory and Examples*. CMS Books in Mathematics (Springer, New York, 2000)Google Scholar - 4.S. Boyd, L. Vandenberghe,
*Convex Optimization*(Cambridge University Press, Cambridge, 2003)zbMATHGoogle Scholar - 5.F.H. Clarke,
*Optimization and Nonsmooth Analysis*(Wiley, New York, 1983)zbMATHGoogle Scholar - 6.R.L. Duncan, Hit probabilities for multiple weapons systems. SIAM Rev.
**6**, 111–114 (1964)MathSciNetCrossRefzbMATHGoogle Scholar - 7.A.R. Eckler, S.A. Burr,
*Mathematical Models of Target Coverage and Missile Allocation*. MORS, Heritage Series (1972), (see also http://www.dtic.mil/dtic/tr/fulltext/u2/a953517.pdf) - 8.J. Egeblad, Heuristics for multidimensional packing problems, Ph.D. thesis, Department of Computer Science, University of Copenhagen (2008), (see also http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.229.3169&rep=rep1&type=pdf)
- 9.D.C. Gilliland, Some bombing problems. Am. Math. Mon.
**73**, 713–716 (1966)MathSciNetCrossRefzbMATHGoogle Scholar - 10.F.E. Grubbs, Approximate circular and noncircular offset probabilities of hitting. Oper. Res.
**12**, 51–62 (1964)CrossRefzbMATHGoogle Scholar - 11.T.C. Hales, Cannonballs and honeycombs. Not. AMS
**47**(4), 440–449 (2000)MathSciNetzbMATHGoogle Scholar - 12.Y. Li, H. Akeb, C.M. Li, Greedy algorithms for packing unequal circles. J. Oper. Res. Soc.
**56**(5), 539–548 (2005)CrossRefzbMATHGoogle Scholar - 13.G. Marsaglia, Some problems involving circular and spherical targets. Oper. Res.
**13**, 18–27 (1965)CrossRefGoogle Scholar - 14.E.H. Neville, On the solution of numerical functional equations. Proc. Lond. Math. Soc.
**14**(2), 308–326 (1915)CrossRefzbMATHGoogle Scholar - 15.J.S. Przemieniecki,
*Mathematical Methods in Defense Analyses*. Air Force Institute of Technology, Education Series, 3rd edn. (American Institute of Aeronautics and Astronautics, Reston, 2000). ISBN-13: 978–1563473975, ISBN-10: 1563473976Google Scholar - 16.K. Stephenson, Circle packing: a mathematical tale. Not. AMS
**50**(11), 1376–1388 (2003)MathSciNetzbMATHGoogle Scholar - 17.C. Uhler, S.J. Wright, Packing ellipsoids with overlap. SIAM Rev.
**55**(4), 671–706 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 18.A.J. Wilson, Volume-of-n-dimensional-ellipsoid. Sciencia Acta Xaveriana (SAX)
**1**(1), 101–106 (2010), http://oaji.net/articles/2014/1420-1415594291.pdf