Solution of elementary equations in the Minkowski geometric algebra of complex sets
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Springer 2005
Solution of elementary equations in the Minkowski geometric algebra of complex sets Rida T. Farouki and Chang Yong Han Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA 95616, USA E-mail: {farouki;cyhan}@ucdavis.edu
Received 17 March 2003; accepted 9 September 2003 Communicated by T. Goodman
The solution of elementary equations in the Minkowski geometric algebra of complex sets is addressed. For given circular disks A and B with radii a and b, a solution of the linear equation A ⊗ X = B in an unknown set X exists if and only if a b. When it exists, the solution 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 A ⊗ (⊗n X ) = 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. Keywords: Minkowski geometric algebra, equations in complex sets, maximal solution, logarithmic Gauss map, Cartesian oval, ovals of Cassini
1.
Introduction
The Minkowski geometric algebra of complex sets is concerned with sets of complex numbers that are populated by algebraic combinations of complex values drawn independently from given complex-set operands [6]. Starting with “simple” set operands1 A and B (e.g., circular disks), elementary binary operations such as the Minkowski sum and Minkowski product A ⊕ B = {a + b | a ∈ A and b ∈ B}, A ⊗ B = {a × b | a ∈ A and b ∈ B} 1 Our convention [5,6] is to denote real values by italic characters, complex values by bold characters, and
sets of complex values by upper-case calligraphic characters.
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R.T. Farouki, C.Y. Han / Minkowski geometric algebra
can be exactly evaluated [5,6]. Elementary unary operations, such as the nth Minkowski power ⊗n A and nth Minkowski root ⊗1/n A, defined by ⊗n A = {z1 z2 · · · zn | zi ∈ A for i = 1, . . . , n} and
z1 z2 · · · zn | zi ∈ ⊗1/n A for i = 1, . . . , n = A,
are also amenable to exact closed-form evaluation [3,7]. A logical next stage in developing this geometric algebra of complex sets is the solution of elementary equations in some unknown set X , with known “simple” sets A, B, . . . as coefficients. For example, when A and B are circular disks, we might ask: under what conditions does a set X exist, that satisfies the linear equation A ⊗ X = B,
(1)
and – when such conditions are met – how can we determine X explicitly? As a natural generalization of (1), we may address the nonlinear monomial equation (2) A ⊗ ⊗n X = B, using the solution to (1) and the ability to extract nth Minkowski roots. The treatment of multin
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