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A NOTE ON COMPUTING THE INTERSECTION OF SPHERES IN $\mathbb{R}^{n}$

Part of: Basic linear algebra Distance geometry

Published online by Cambridge University Press:  02 November 2017

D. S. MAIOLI Open the ORCID record for D. S. MAIOLI [Opens in a new window] ,
C. LAVOR  and
D. S. GONÇALVES Open the ORCID record for D. S. GONÇALVES [Opens in a new window]
D. S. MAIOLI*
Affiliation:
Department of Applied Mathematics, IMECC, University of Campinas, Campinas, Brazil email ra099812@ime.unicamp.br, clavor@ime.unicamp.br
C. LAVOR
Affiliation:
Department of Applied Mathematics, IMECC, University of Campinas, Campinas, Brazil email ra099812@ime.unicamp.br, clavor@ime.unicamp.br
D. S. GONÇALVES
Affiliation:
Department of Mathematics, CCFM, Federal University of Santa Catarina, Florianópolis, Brazil email douglas.goncalves@ufsc.br
*
✉ra099812@ime.unicamp.br
Rights & Permissions [Opens in a new window]

Abstract

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Finding the intersection of $n$-dimensional spheres in $\mathbb{R}^{n}$ is an interesting problem with applications in trilateration, global positioning systems, multidimensional scaling and distance geometry. In this paper, we generalize some known results on finding the intersection of spheres, based on QR decomposition. Our main result describes the intersection of any number of $n$-dimensional spheres without the assumption that the centres of the spheres are affinely independent. A possible application in the interval distance geometry problem is also briefly discussed.

Keywords

sphere intersectionn-dimensional spheresQR decomposition

MSC classification

Primary: 51K05: General theory
Secondary: 15A03: Vector spaces, linear dependence, rank
Type
Research Article
Information
The ANZIAM Journal , Volume 59 , Issue 2 , October 2017 , pp. 271 - 279
Copyright
© 2017 Australian Mathematical Society 

References

Alves, R. and Lavor, C., “Geometric algebra to model uncertainties in the discretizable molecular distance geometry problem”, Adv. Appl. Clifford Algebr. 27 (2017) 439–452; doi:10.1007/s00006-016-0653-2.CrossRefGoogle Scholar
Billinge, S., Duxbury, P., Gonçalves, D. S., Lavor, C. and Mucherino, A., “Assigned and unassigned distance geometry: applications to biological molecules and nanostructures”, 4OR – Q. J. Oper. Res. 14 (2016) 337–376; doi:10.1007/s10288-016-0314-2.CrossRefGoogle Scholar
Biswas, P., Liang, T., Wang, T. and Ye, Y., “Semidefinite programming based algorithms for sensor network localization”, ACM Trans. Sens. Netw. 2 (2006) 188–220; doi:10.1145/1149283.1149286.Google Scholar
Blumenthal, L., Theory and applications of distance geometry (Clarendon Press, Oxford, 1953).Google Scholar
Borg, I. and Groenen, P., Modern multidimensional scaling, 2nd edn. (Springer, New York, 2010); doi:10.1007/978-1-4757-2711-1.Google Scholar
Cassioli, A., Bardiaux, B., Bouvier, G., Mucherino, A., Alves, R., Liberti, L., Nilges, M., Lavor, C. and Malliavin, T., “An algorithm to enumerate all possible protein conformations verifying a set of distance constraints”, BMC Bioinform. 16 (2015) 16–23; doi:10.1186/s12859-015-0451-1.CrossRefGoogle ScholarPubMed
Coope, I., “Reliable computation of the points of intersection of $n$ spheres in $\mathbb{R}^{n}$ ”, ANZIAM J. 42 (2000) 461–477; doi:10.21914/anziamj.v42i0.608.Google Scholar
Crippen, G. and Havel, T., Distance geometry and molecular conformation (John Wiley and Sons, New York, 1988).Google Scholar
Dong, Q. and Wu, Z., “A geometric build-up algorithm for solving the molecular distance geometry problem with sparse distance data”, J. Global Optim. 26 (2003) 321–333; doi:10.1023/A:1023221624213.Google Scholar
Golub, G. H. and Van Loan, C. F., Matrix computations, 3rd edn. (JHU Press, Baltimore, MD, 1996).Google Scholar
Gonçalves, D. and Mucherino, A., “Discretization orders and efficient computation of Cartesian coordinates for distance geometry”, Optim. Lett. 8 (2014) 2111–2125; doi:10.1007/s11590-014-0724-z.Google Scholar
Gonçalves, D., Mucherino, A., Lavor, C. and Liberti, L., “Recent advances on the interval distance geometry problem”, J. Global Optim. (2017) 1–21; doi:10.1007/s10898-016-0493-6.Google Scholar
Lavor, C., Alves, R., Figueiredo, W., Petraglia, A. and Maculan, N., “Clifford algebra and the discretizable molecular distance geometry problem”, Adv. Appl. Clifford Algebr. 25 (2015) 925–942; doi:10.1007/s00006-015-0532-2.CrossRefGoogle Scholar
Lavor, C., Liberti, L., Maculan, N. and Mucherino, A., “Recent advances on the discretizable molecular distance geometry problem”, Eur. J. Oper. Res. 219 (2012) 698–706; doi:10.1016/j.ejor.2011.11.007.CrossRefGoogle Scholar
Lavor, C., Liberti, L., Maculan, N. and Mucherino, A., “The discretizable molecular distance geometry problem”, Comput. Optim. Appl. 52 (2012) 115–146; doi:10.1007/s10589-011-9402-6.Google Scholar
Lavor, C., Liberti, L. and Mucherino, A., “The interval branch-and-prune algorithm for the discretizable molecular distance geometry problem with inexact distances”, J. Global Optim. 56 (2013) 855–871; doi:10.1007/s10898-011-9799-6.Google Scholar
Liberti, L. and Lavor, C., “Six mathematical gems from the history of distance geometry”, Int. Trans. Oper. Res. 23 (2016) 897–920; doi:10.1111/itor.12170.CrossRefGoogle Scholar
Liberti, L., Lavor, C. and Maculan, N., “A branch-and-prune algorithm for the molecular distance geometry problem”, Int. Trans. Oper. Res. 15 (2008) 1–17; doi:10.1111/j.1475-3995.2007.00622.x.Google Scholar
Liberti, L., Lavor, C., Maculan, N. and Mucherino, A., “Euclidean distance geometry and applications”, SIAM Rev. 56 (2014) 3–69; doi:10.1137/120875909.Google Scholar
Mucherino, A., Lavor, C., Liberti, L. and Maculan, N., Distance geometry: theory, methods and application (Springer, New York, NY, 2013); doi:10.1007/978-1-4614-5128-0.Google Scholar
Nielsen, J. and Roth, B., “On the kinematic analysis of robotic mechanisms”, Int. J. Robot. Res. 18 (1999) 1147–1160; doi:10.1177/02783649922067771.Google Scholar
Wu, D. and Wu, Z., “An updated geometric build-up algorithm for solving the molecular distance geometry problem with sparse distance data”, J. Global Optim. 37 (2007) 661–673; doi:10.1007/s10898-006-9080-6.CrossRefGoogle Scholar