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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 [email protected], [email protected]
C. LAVOR
Affiliation:
Department of Applied Mathematics, IMECC, University of Campinas, Campinas, Brazil email [email protected], [email protected]
D. S. GONÇALVES
Affiliation:
Department of Mathematics, CCFM, Federal University of Santa Catarina, Florianópolis, Brazil email [email protected]
*
✉[email protected]
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 

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