Skip to main content
SpringerLink
Log in

Minkowski Geometric Algebra of Complex Sets

  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

A geometric algebra of point sets in the complex plane is proposed, based on two fundamental operations: Minkowski sums and products. Although the (vector) Minkowski sum is widely known, the Minkowski product of two-dimensional sets (induced by the multiplication rule for complex numbers) has not previously attracted much attention. Many interesting applications, interpretations, and connections arise from the geometric algebra based on these operations. Minkowski products with lines and circles are intimately related to problems of wavefront reflection or refraction in geometrical optics. The Minkowski algebra is also the natural extension, to complex numbers, of interval-arithmetic methods for monitoring propagation of errors or uncertainties in real-number computations. The Minkowski sums and products offer basic 'shape operators' for applications such as computer-aided design and mathematical morphology, and may also prove useful in other contexts where complex variables play a fundamental role – Fourier analysis, conformal mapping, stability of control systems, etc.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Artin, E.: Geometric Algebra, Wiley, New York, 1957.

    Google Scholar 

  2. Bernoulli, J.: Lineæ cycloidales, evolutæ, ant-evolutæ, causticæ, anti-causticæ, pericausticæ, Acta Eruditorum, May 1692.

  3. Boltyanskii, V. G.: Envelopes, Macmillan, New York, 1964.

    Google Scholar 

  4. Bruce, J. W. and Giblin, P. J.: What is an envelope?, Math. Gazette 65 (1981), 186–192.

    Google Scholar 

  5. Bruce, J. W. and Giblin, P. J.: Curves and Singularities, Cambridge Univ. Press, 1984.

  6. Cayley, A.: A memoir upon caustics, Phil. Trans. Roy. Soc. London 147, 273–312; Supplementary memoir upon caustics, Phil. Trans. Roy. Soc. London 157 (1857), 7–16.

  7. Chastang, J-C. A. and Farouki, R. T.: The mathematical evolution of wavefronts, Optics Photonics News 3 (1992), 20–23.

    Google Scholar 

  8. Choi, H. I., Han, C. Y., Moon, H. P., Roh, K. H. and Wee, N-S.: Medial axis transform and offset curves by Minkowski Pythagorean hodograph curves, Comput. Aided Design 31 (1999), 59–72.

    Google Scholar 

  9. Deaux, R.: Introduction to the Geometry of Complex Numbers (translated from the French by H. Eves), F. Ungar, New York, 1956.

    Google Scholar 

  10. Dorf, R. C.: Modern Control Systems (2nd edn), Addison-Wesley, Reading. MA, 1974.

    Google Scholar 

  11. Elber, G., Lee, I-K. and Kim, M-S.: Comparing offset curve approximation methods, IEEE Comput. Graphics Appl. 17(3) (1997), 62–71.

    Google Scholar 

  12. Farin, G.: Curves and Surfaces for CAGD (3rd edn), Academic Press, Boston, 1993.

    Google Scholar 

  13. Farouki, R. T.: The conformal map z → z 2 of the hodograph plane, Comput. Aided Geom. Design 11 (1994), 363–390.

    Google Scholar 

  14. Farouki, R. T. and Chastang, J-C. A.: Curves and surfaces in geometrical optics, In: T. Lyche and L. L. Schumaker, (eds), Mathematical Methods in Computer Aided Geometric Design II, Academic Press, New York, pp. 239–260.

  15. Farouki, R. T. and Chastang, J-C. A.: Exact equations of `simple' wavefronts, Optik 91 (1992), 109–121.

    Google Scholar 

  16. Farouki, R. T. and Johnstone, J. K.: Computing point/curve and curve/curve bisectors, In: R. B. Fisher,(ed), Design and Application of Curves and Surfaces (The Mathematics of Surfaces V) Oxford Univ. Press, 1994, pp. 327–354.

  17. Farouki, R. T., Moon, H. P. and Ravani, B.: Algorithms for Minkowski products and implicitly-defined complex sets, Adv. Comput. Math. 13 (2000), 199–229.

    Google Scholar 

  18. Farouki, R. T. and Neff, C. A.: Analytic properties of plane offset curves & Algebraic properties of plane offset curves, Comput. Aided Geom. Design 7 (1990), 83–99, 101–127.

    Google Scholar 

  19. Farouki, R. T. and Neff, C. A.: Hermite interpolation by Pythagorean-hodograph quintics, Math. Comput. 64 (1995), 1589–1609.

    Google Scholar 

  20. Farouki, R. T. and Sakkalis, T.: Pythagorean hodographs, IBM J. Res. Develop. 34 (1990), 736–752.

    Google Scholar 

  21. Fowler, R. H.: The Elementary Differential Geometry of Plane Curves, Cambridge Univ. Press, 1929.

  22. Ghosh, P. K.: A mathematical model for shape description using Minkowski operators, Comput. Vision, Graphics, Image Process. 44 (1988), 239–269.

    Google Scholar 

  23. Gomes Teixeira, F.: Traitédes courbes Spéciales Remarquables planes et gauches, Tome I, Chelsea (reprint),New York, 1971.

  24. Hadwiger, H.: Vorlesungen über Inhalt, Oberfl#x00c4;che, und Isoperimetrie, Springer, Berlin, 1957.

    Google Scholar 

  25. Hansen, E.: Interval forms of Newton's method, Computing 20 (1978), 153–163.

    Google Scholar 

  26. Hansen, E.: A globally convergent interval method for computing and bounding real roots, BIT 18 (1978), 415–424.

    Google Scholar 

  27. Hartquist, E. E., Menon, J. P., Suresh, K., Voelcker, H. B. and Zagajac, J.: A computing strategy for applications involving offsets, sweeps, and Minkowski operators, Comput. Aided Design 31 (1999), 175–183.

    Google Scholar 

  28. Held, M.: On the Computational Geometry of Pocket Machining, Springer-Verlag, Berlin, 1991.

    Google Scholar 

  29. Herman, R. A.: A Treatise on Geometrical Optics, Cambridge Univ. Press, 1900.

  30. Huppert, B.: Geometric Algebra, Lecture notes, Mathematics Department, University of Illinois at Chicago, 1970.

    Google Scholar 

  31. Jacobs, O. L. R.: Introduction to Control Theory, Clarendon Press, Oxford, 1974.

    Google Scholar 

  32. Kaul, A.: Computing Minkowski sums, PhD Thesis, Columbia University, 1993.

  33. Kaul, A. and Farouki, R. T. Computing Minkowski sums of plane curves, Int. J. Comput. Geom. Applic. 5 (1995), 413–432.

    Google Scholar 

  34. Kaul, A. and Rossignac, J. R.: Solid interpolating deformations: Construction and animation of PIP, Computers and Graphics 16 (1992), 107–115.

    Google Scholar 

  35. Lin, Q. and Rokne, J.: Disk Bézier curves, Comput. Aided Geom. Design 15 (1998), 721–737.

    Google Scholar 

  36. Lawrence, J. D.: A Catalog of Special Plane Curves, Dover, New York, 1972.

  37. Lockwood, E. H.: A Book of Curves, Cambridge Univ. Press, 1967.

  38. Lozano-Pérez, T. and Wesley, M. A.: An algorithm for planning collision-free paths among polyhedral obstacles, Comm. ACM 22 (1979), 560–570.

    Google Scholar 

  39. Matheron, G.: Random Sets and Integral Geometry, Wiley, New York, 1976

    Google Scholar 

  40. McDonald, B. R.: Geometric Algebra over Local Rings, M. Dekker, New York, 1976.

    Google Scholar 

  41. Middleditch, A. E.: Applications of a vector sum operator, Comput. Aided Design 20 (1988), 183–188.

    Google Scholar 

  42. Minkowski, H.: Volumen und Oberfläche, Math. Ann. 57 (1903), 447–495.

    Google Scholar 

  43. Moon, H. P.: Minkowski Pythagorean hodographs, Comput. Aided Geom. Design 16 (1999), 739–753.

    Google Scholar 

  44. Moore, R. E.: Interval Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1966.

    Google Scholar 

  45. Moore, R. E.: Methods and Applications of Interval Analysis, SIAM, Philadelphia, 1979.

    Google Scholar 

  46. Mudur, S. P. and Koparkar, P. A.: Interval methods for processing geometric objects, IEEE Comput. Graphics Appl. 2 (1984), 7–17.

    Google Scholar 

  47. Needham, T.: Visual Complex Analysis, Oxford Univ. Press, 1997.

  48. Pottmann, H.: Rational curves and surfaces with rational offsets, Comput. Aided Geom. Design 12 (1995), 175–192.

    Google Scholar 

  49. Ramamurthy, R. and Farouki, R. T.: Voronoi diagram and medial axis algorithm for planar domains with curved boundaries I. Theoretical foundations & II. Detailed algorithm description, J. Comput. Appled Math. 102 (1999), 119–141, 253–277.

    Google Scholar 

  50. Rossignac, J. R. and Requicha, A. A. G.: Offsetting operations in solid modelling, Comput. Aided Geom. Design 3 (1986), 129–148.

    Google Scholar 

  51. Salmon, G.: A Treatise on the Higher Plane Curves, Chelsea, New York (reprint), 1960.

    Google Scholar 

  52. Schwerdtfeger, H.: Geometry of Complex Numbers, Dover, New York, 1979.

  53. Sederberg, T. W. and Farouki, R. T.: Approximation by interval Bézier curves, IEEE Comput. Graphics Applic. 126(5) (1992), 87–95.

    Google Scholar 

  54. Serra, J.: Image Analysis and Mathematical Morphology, Academic Press, London, 1982.

    Google Scholar 

  55. Serra, J.: Introduction to mathematical morphology, Comput. Vision, Graphics, Image Process. 35 (1986), 283–305.

    Google Scholar 

  56. Stavroudis, O. N.: The Optics of Rays, Wavefronts, and Caustics, Academic Press, New York, 1972.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA, 95616, U.S.A.

    Rida T. Farouki

  2. Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA, 95616, U.S.A.

    Hwan Pyo Moon

  3. Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA, 95616, U.S.A.

    Bahram Ravani

Authors
  1. Rida T. Farouki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hwan Pyo Moon
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bahram Ravani
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Farouki, R.T., Moon, H.P. & Ravani, B. Minkowski Geometric Algebra of Complex Sets. Geometriae Dedicata 85, 283–315 (2001). https://doi.org/10.1023/A:1010318011860

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1010318011860

Search

Navigation