Abstract
Minkowski geometric algebra is concerned with the complex sets populated by the sums and products of all pairs of complex numbers selected from given complex‐set operands. Whereas Minkowski sums (under vector addition in Rn have been extensively studied, from both the theoretical and computational perspective, Minkowski products in R2 (induced by the multiplication of complex numbers) have remained relatively unexplored. The complex logarithm reveals a close relation between Minkowski sums and products, thereby allowing algorithms for the latter to be derived through natural adaptations of those for the former. A novel concept, the logarithmic Gauss maps of plane curves, plays a key role in this process, furnishing geometrical insights that parallel those associated with the “ordinary” Gauss map. As a natural generalization of Minkowski sums and products, the computation of “implicitly‐defined” complex sets (populated by general functions of values drawn from given sets) is also considered. By interpreting them as one‐parameter families of curves, whose envelopes contain the set boundaries, algorithms for evaluating such sets are sketched.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J.W. Bruce and P.J. Giblin, Curves and Singularities, 2nd ed. (Cambridge Univ. Press, Cambridge, 1992).
R. Deaux, Introduction to the Geometry of Complex Numbers (translated from the French by H. Eves) (F. Ungar, New York, 1956).
M.P. do Carmo, Differential Geometry of Curves and Surfaces (Prentice-Hall, Englewood Cliffs, NJ, 1976).
G. Farin, Curves and Surfaces for CAGD, 3rd ed. (Academic Press, Boston, 1993).
R.T. Farouki and T.N.T. Goodman, On the optimal stability of the Bernstein basis, Math. Comp. 65 (1996) 1553–1566.
R.T. Farouki, H.P. Moon and B. Ravani, Minkowski geometric algebra of complex sets, Geom. Dedicata (2000) to appear.
R.T. Farouki and C.A. Neff, Analytic properties of plane offset curves & Algebraic properties of plane offset curves, Comput. Aided Geom. Design 7(1990) 83–99 & 101?127.
R.T. Farouki and C.A. Neff, Hermite interpolation by Pythagorean-hodograph quintics, Math. Comp. 64 (1995) 1589–1609.
R.T. Farouki and V.T. Rajan, On the numerical condition of polynomials in Bernstein form, Comput. Aided Geom. Design 4(1987) 191–216.
P.K. Ghosh, A mathematical model for shape description using Minkowski operators, Comput. Vision Graphics Image Process. 44 (1988) 239–269.
H. Hadwiger, Vorlesungen über Inhalt, Oberfläche, und Isoperimetrie (Springer, Berlin, 1957).
A. Kaul, Computing Minkowski sums, Ph.D. thesis, Columbia University (1993).
A. Kaul and R.T. Farouki, Computing Minkowski sums of plane curves, Internat. J. Comput. Geom. Appl. 5 (1995) 413–432.
J.M. Lane and R.F. Riesenfeld, Bounds on a polynomial, BIT 21(1981) 112–117.
J.D. Lawrence, A Catalog of Special Plane Curves (Dover, New York, 1972).
I.K. Lee and M.S. Kim, Polynomial/rational approximation of Minkowski sum boundary curves, Graphical Models Image Process. 60 (1998) 136–165.
I.K. Lee, M.S. Kim, and G. Elber, New approximation methods of planar offset and convolution curves, in: Geometric Modeling: Theory and Practice, eds. W. Strasser, R. Klein and R. Rau (Springer, Berlin, 1997) pp. 83–101.
E.H. Lockwood, A Book of Curves (Cambridge Univ. Press, Cambridge, 1967).
H. Minkowski, Volumen und Oberfläche, Math. Ann. 57 (1903) 447–495.
R.E. Moore, Interval Analysis (Prentice-Hall, Englewood Cliffs, NJ, 1966).
R.E. Moore, Methods and Applications of Interval Analysis (SIAM, Philadelphia, PA, 1979).
T. Needham, Visual Complex Analysis (Oxford Univ. Press, Oxford, 1997).
R. Schneider, Convex Bodies: The Brunn-Minkowski Theory (Cambridge Univ. Press, Cambridge, 1993).
H. Schwerdtfeger, Geometry of Complex Numbers (Dover, New York, 1979).
T.W. Sederberg and S.R. Parry, Comparison of three curve intersection algorithms, Comput. Aided Design 18 (1986) 58–64.
J. Serra, Image Analysis and Mathematical Morphology (Academic Press, London, 1982).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Farouki, R.T., Moon, H.P. & Ravani, B. Algorithms for Minkowski products and implicitly‐defined complex sets. Advances in Computational Mathematics 13, 199–229 (2000). https://doi.org/10.1023/A:1018910412112
Issue Date:
DOI: https://doi.org/10.1023/A:1018910412112