Abstract
An exact parameterization for the boundary of the Minkowski product of N circular disks in the complex plane is derived. When N > 2, this boundary curve may be regarded as a generalization of the Cartesian oval that bounds the Minkowski product of two disks. The derivation is based on choosing a system of coordinated polar representations for the N operands, identifying sets of corresponding points with matched logarithmic Gauss map that may contribute to the Minkowski product boundary. By means of inversion in the operand circles, a geometrical characterization for their corresponding points is derived, in terms of intersections with the circles of a special coaxal system. The resulting parameterization is expressed as a product of N terms, each involving the radius of one disk, a single square root, and the sine and cosine of a common angular variable ϕ over a prescribed domain. As a special case, the N-th Minkowski power of a single disk is bounded by a higher trochoid. In certain applications, the availability of exact Minkowski products is a useful alternative to the naive bounding approximations that are customarily employed in "complex circular arithmetic."
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alefeld, G. and Herzberger, J.: Introduction to Interval Computations, Academic Press, New York, 1983.
Boltyanskii, V. G.: Envelopes, Macmillan, New York, 1964.
Brannan, D. A., Esplen, M. F., and Gray, J. J.: Geometry, Cambridge University Press, 1999.
Bruce, J. W. and Giblin, P. J.: Curves and Singularities, Cambridge University Press, 1984.
Chapellat, H., Bhattacharyya, S. P., and Dahleh, M.: Robust Stability of a Family of Disc Polynomials, International Journal of Control 51 (1990), pp. 1353–1362.
Coolidge, J. L.: A Treatise on the Circle and the Sphere, Clarendon Press, Oxford, 1916.
Farouki, R. T. and Chastang, J.-C. A.: Curves and Surfaces in Geometrical Optics, in: Lyche, T. and Schumaker, L. L. (eds), Mathematical Methods in Computer Aided Geometric Design II, Academic Press, 1992, pp. 239–260.
Farouki, R. T. and Chastang, J.-C. A.: Exact Equations of “Simple”Wavefronts, Optik 91 (1992), pp. 109–121.
Farouki, R. T., Gu, W., and Moon, H. P.: Minkowski Roots of Complex Sets, in: Geometric Modeling and Processing 2000, IEEE Computer Society Press, 2000, pp. 287–300.
Farouki, R. T. and Moon, H. P.: Bipolar and Multipolar Coordinates, in: Cippola, R. (ed.), The Mathematics of Surfaces IX, Springer, 2000, pp. 348–371.
Farouki, R. T., Moon, H. P., and Ravani, B.: Algorithms for Minkowski Products and Implicitly-Defined Complex Sets, Advances in Computational Mathematics 13 (2000), pp. 199–229.
Farouki, R. T., Moon, H. P., and Ravani, B.: Minkowski Geometric Algebra of Complex Sets, Geometriae Dedicata 85 (2001), pp. 283–315.
Gargantini, I. and Henrici, P.: Circular Arithmetic and the Determination of Polynomial Zeros, Numerische Mathematik 18 (1972), pp. 305–320.
Gomes Teixeira, F.: Traité des Courbes Spéciales Remarquables Planes et Gauches, Tome I, Chelsea (reprint), New York, 1971.
Hahn, L.-S.: Complex Numbers and Geometry, Mathematical Association of America, Washington, D.C., 1994.
Hauenschild, M.: Arithmetiken für komplexe Kreise, Computing 13 (1974), pp. 299–312.
Hauenschild, M.: Extended Circular Arithmetic, Problems and Results, in: Nickel, K. L. E. (ed.), Interval Mathematics 1980, Academic Press, New York, (1980), pp. 367–376.
Henrici, P.: Applied and Computational Complex Analysis, Vol. I, Wiley, New York, 1974.
Lawrence, J. D.: A Catalog of Special Plane Curves, Dover, New York, 1972.
Lin, Q. and Rokne, J. G.: Disk Bézier Curves, Computer Aided Geometric Design 15 (1998), pp. 721–737.
Lockwood, E. H.: A Book of Curves, Cambridge University Press, 1967.
Moore, R. E.: Interval Analysis, Prentice Hall, Englewood Cliffs, 1966.
Moore, R. E.: Methods and Applications of Interval Analysis, SIAM, Philadelphia, 1979.
Needham, T.: Visual Complex Analysis, Oxford University Press, 1997.
Pedoe, D.: Geometry: A Comprehensive Course, Dover, New York, 1970, reprint.
Petković, M. S. and Petković, L. D.: Complex Interval Arithmetic and Its Applications, Wiley-VCH, Berlin, 1998.
Polyak, B. T., Scherbakov, P. S., and Shmulyian, S. B.: Construction of Value Set for Robustness Analysis via Circular Arithmetic, International Journal of Robust and Nonlinear Control 4 (1994), pp. 371–385.
Pottmann, H.: Rational Curves and Surfaces with Rational Offsets, Computer Aided Geometric Design 12 (1995), pp. 175–192.
Ratschek, H. and Rokne, J.: Computer Methods for the Range of Functions, Ellis Horwood, Chichester, 1984.
Schwerdtfeger, H.: Geometry of Complex Numbers, Dover, New York, 1979.
Wunderlich, W.: Höhere Radlinien, Österreichisches Ingenieur-Archiv 1 (1947), pp. 277–296.
Zwikker, C.: The Advanced Geometry of Plane Curves and Their Applications, Dover, New York, 1963.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Farouki, R.T., Pottmann, H. Exact Minkowski Products of N Complex Disks. Reliable Computing 8, 43–66 (2002). https://doi.org/10.1023/A:1014737602641
Issue Date:
DOI: https://doi.org/10.1023/A:1014737602641