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."
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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.
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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
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DOI: https://doi.org/10.1023/A:1014737602641