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Advances in Computational Mathematics

, Volume 45, Issue 3, pp 1607–1629 | Cite as

Minkowski products of unit quaternion sets

  • Rida T. FaroukiEmail author
  • Graziano Gentili
  • Hwan Pyo Moon
  • Caterina Stoppato
Article

Abstract

The Minkowski product of unit quaternion sets is introduced and analyzed, motivated by the desire to characterize the overall variation of compounded spatial rotations that result from individual rotations subject to known uncertainties in their rotation axes and angles. For a special type of unit quaternion set, the spherical caps of the 3-sphere S3 in \(\mathbb {R}^{4}\), closure under the Minkowski product is achieved. Products of sets characterized by fixing either the rotation axis or rotation angle, and allowing the other to vary over a given domain, are also analyzed. Two methods for visualizing unit quaternion sets and their Minkowski products in \(\mathbb {R}^{3}\) are also discussed, based on stereographic projection and the Lie algebra formulation. Finally, some general principles for identifying Minkowski product boundary points are discussed in the case of full-dimension set operands.

Keywords

Minkowski products Unit quaternions Spatial rotations 3-sphere Stereographic projection Lie algebra Boundary evaluation 

Mathematics Subject Classification (2010)

65G30 30G35 

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Notes

Funding information

The second and fourth authors are partly supported by the Italian Ministry of Education (MIUR) through Finanziamento Premiale FOE 2014 “Splines for accUrate NumeRics: adaptIve models for Simulation Environments” and by Istituto Nazionale di Alta Matematica (INdAM) through Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni (GNSAGA).

References

  1. 1.
    Altmann, S.L.: Rotations, Quaternions, and Double Groups. Dover Publications (reprint), Mineola (1986)zbMATHGoogle Scholar
  2. 2.
    Du Val, P.: Homographies, Quaternions, and Rotations. Clarendon Press, Oxford (1964)zbMATHGoogle Scholar
  3. 3.
    Farouki, R.T., Gu, W., Moon, H.P.: Minkowski roots of complex sets. In: Geometric Modeling and Processing 2000, pp. 287–300. IEEE Computer Society Press (2000)Google Scholar
  4. 4.
    Farouki, R.T., Han, C.Y.: Computation of Minkowski values of polynomials over complex sets. Numer. Algor. 36, 13–29 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Farouki, R.T., Han, C.Y.: Solution of elementary equations in the Minkowski geometric algebra of complex sets. Adv. Comput. Math. 22, 301–323 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Farouki, R.T., Moon, H.P., Ravani, B.: Algorithms for Minkowski products and implicitly–defined complex sets. Adv. Comput. Math. 13, 199–229 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Farouki, R.T., Moon, H.P., Ravani, B.: Minkowski geometric algebra of complex sets. Geom. Dedicata. 85, 283–315 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Farouki, R.T., Pottmann, H.: Exact Minkowski products of N complex disks. Reliab. Comput. 8, 43–66 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ghosh, P.K.: A mathematical model for shape description using Minkowski operators. Comput. Vis. Graph. Image Process. 44, 239–269 (1988)CrossRefGoogle Scholar
  10. 10.
    Gilmore, R.: Lie Groups, Physics, and Geometry: An Introduction for Physicists, Engineers and Chemists. Cambridge University Press, Cambridge (2008)CrossRefzbMATHGoogle Scholar
  11. 11.
    Hartquist, E.E., Menon, J.P., Suresh, K., Voelcker, H.B., Zagajac, J.: A computing strategy for applications involving offsets, sweeps, and Minkowski operators. Comput. Aided Des. 31, 175–183 (1999)CrossRefzbMATHGoogle Scholar
  12. 12.
    Kaul, A.: Computing Minkowski Sums. PhD Thesis, Columbia University (1993)Google Scholar
  13. 13.
    Kaul, A., Farouki, R.T.: Computing Minkowski sums of plane curves. Int. J. Comput. Geom. Appl. 5, 413–432 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kaul, A., Rossignac, J.R.: Solid interpolating deformations: Construction and animation of PIP. Comput. Graph. 16, 107–115 (1992)CrossRefGoogle Scholar
  15. 15.
    Middleditch, A.E.: Applications of a vector sum operator. Comput. Aided Des. 20, 183–188 (1988)CrossRefzbMATHGoogle Scholar
  16. 16.
    Minkowski, H.: Volumen und Oberfläche. Math. Ann. 57, 447–495 (1903)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Moore, R.E.: Interval Analysis. Prentice–Hall, Englewood Cliffs (1966)Google Scholar
  18. 18.
    Moore, R.E: Methods and Applications of Interval Analysis. SIAM, Philadelphia (1979)CrossRefzbMATHGoogle Scholar
  19. 19.
    Selig, J.M.: Geometrical Methods in Robotics. Springer, New York (1996)CrossRefzbMATHGoogle Scholar
  20. 20.
    Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, London (1982)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Rida T. Farouki
    • 1
    Email author
  • Graziano Gentili
    • 2
  • Hwan Pyo Moon
    • 3
  • Caterina Stoppato
    • 2
  1. 1.Department of Mechanical and Aerospace EngineeringUniversity of CaliforniaDavisUSA
  2. 2.Dipartimento di Matematica e Informatica “U. Dini,”Università di FirenzeFirenzeItaly
  3. 3.Department of MathematicsDongguk University–SeoulSeoulRepublic of Korea

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