Abstract
The solution of elementary equations in the Minkowski geometric algebra of complex sets is addressed. For given circular disks \(\mathcal{A}\) and ℬ with radii a and b, a solution of the linear equation \(\mathcal{A}\otimes \mathcal{X}=\mathcal{B}\) in an unknown set \(\mathcal{X}\) exists if and only if a≤b. When it exists, the solution \(\mathcal{X}\) is generically the region bounded by the inner loop of a Cartesian oval (which may specialize to a limaçon of Pascal, an ellipse, a line segment, or a single point in certain degenerate cases). Furthermore, when a<b<1, the solution of the nonlinear monomial equation \(\mathcal{A}\otimes(\otimes^{n}\mathcal{X})=\mathcal{B}\) is shown to be the region that is bounded by a single loop of a generalized form of the ovals of Cassini. The latter result is obtained by considering the nth Minkowski root of the region bounded by the inner loop of a Cartesian oval. Preliminary consideration is also given to the problems of solving univariate polynomial equations and multivariate linear equations with complex disk coefficients.
Similar content being viewed by others
References
R.T. Farouki and J.-C.A. Chastang, Curves and surfaces in geometrical optics, in: Mathematical Methods in Computer Aided Geometric Design, Vol. II, eds. T. Lyche and L.L. Schumaker (Academic Press, New York, 1992) pp. 239–260.
R.T. Farouki and J.-C.A. Chastang, Exact equations of “simple” wavefronts, Optik 91 (1992) 109–121.
R.T. Farouki, W. Gu and H.P. Moon, Minkowski roots of complex sets, in: Geometric Modeling and Processing 2000 (IEEE Computer Soc. Press, Los Alamitos, CA, 2000) pp. 287–300.
R.T. Farouki and H.P. Moon, Minkowski geometric algebra and the stability of characteristic polynomials, in: Visualization and Mathematics, Vol. 3, eds. H.-C. Hege and K. Polthier (Springer, Berlin, 2003) pp. 163–188.
R.T. Farouki, H.P. Moon and B. Ravani, Algorithms for Minkowski products and implicitly-defined complex sets, Adv. Comput. Math. 13 (2000) 199–229.
R.T. Farouki, H.P. Moon and B. Ravani, Minkowski geometric algebra of complex sets, Geometriae Dedicata 85 (2001) 283–315.
R.T. Farouki and H. Pottmann, Exact Minkowski products of N complex disks, Reliable Comput. 8 (2002) 43–66.
I. Gargantini and P. Henrici, Circular arithmetic and the determination of polynomial zeros, Numer. Math. 18 (1972) 305–320.
F. Gomes Teixeira, Traité des Courbes Spéciales Remarquables Planes et Gauches, Tome I (reprint) (Chelsea, New York, 1971).
M. Hauenschild, Arithmetiken für komplexe Kreise, Computing 13 (1974) 299–312.
M. Hauenschild, Extended circular arithmetic, problems and results, in: Interval Mathematics 1980, ed. K.L.E. Nickel (Academic Press, New York, 1980) pp. 367–376.
J.D. Lawrence, A Catalog of Special Plane Curves (Dover, New York, 1972).
E.H. Lockwood, A Book of Curves (Cambridge Univ. Press, Cambridge, 1967).
R.E. Moore, Interval Analysis (Prentice-Hall, Englewood Cliffs, NJ, 1966).
R.E. Moore, Methods and Applications of Interval Analysis (SIAM, Philadelphia, PA, 1979).
M.S. Petković and L.D. Petković, Complex Interval Arithmetic and Its Applications (Wiley/VCH, Berlin, 1998).
Author information
Authors and Affiliations
Additional information
Communicated by T. Goodman
Rights and permissions
About this article
Cite this article
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). https://doi.org/10.1007/s10444-003-2605-3
Issue Date:
DOI: https://doi.org/10.1007/s10444-003-2605-3