Minkowski Geometric Algebra of Complex Sets: Theory, Algorithms, and Applications

摘要：加州伯克利minkowski几何代数的rida FaroukiU关注的是由给定复Cset操作数所得到的所有对复数的和与积所填充的复集合。虽然rn中向量相加下的闵可夫斯基和已经得到了深入的研究，但从理论和计算的角度来看，由复数乘法规则导出的r2中的闵可夫斯基积是一个相对未被探索的概念。从概念上讲，闵可夫斯基几何代数是实区间算术对复数集的自然推广。然而，随着实数到复数的过渡，实数区间的平凡几何及其在加法和乘法下的闭合被放弃了。闵可夫斯基几何代数的二维特征赋予了它丰富的几何内容，其中简单的操作数（如圆盘）产生由解析曲线描述的微妙结果，如卡西尼的笛卡尔椭圆及其高阶推广，而一般集合的闵可夫斯基组合的近似需要复杂的算法。除了作为监视复杂c变量计算中不确定性传播的基本工具之外，代数o。为二维c维形状算子和（正或逆）波前re的描述提供了一种通用语言。光学中的反射/折射问题。在Routh CHurwitz准则和Kharitonov鲁棒性定理的背景下，它也是具有不确定参数的系统稳定性分析的基础。

ABSTRACT0202179Rida FaroukiU of Calif BerkeleyMinkowski geometric algebra is concerned with the complex sets populated by the sums and products of all pairs of complex numbers drawn from given complex Cset operands.Whereas the Minkowski sum under vector addition in R n has been thoroughly studied,from both the theoretical and computationalperspective,the Minkowski product in R 2 induced by the multiplication rule for complex numbers is a relatively unexplored oncept.Conceptually, Minkowski geometric algebra is the natural generalization of real interval arithmetic to complex Cnumber sets.With the transition from real to complex,however,the trivial geometry of real intervals and their consequent closure under addition and multiplication is relinquished. Thetwo Cdimensional haracter of Minkowski geometric algebra endows it with a rich geometrical ontent,in which simple operands (e.g.,circular disks) yield subtle results described by analytic curves such as the Cartesian oval ovals of Cassini and their higher Corder generalizations,while sophisticated algorithms are required to approximate Minkowski combinations for general sets.Apart from being a basic tool to monitor the propagation of uncertainty in complex Cvariable computations,the algebra o .ers a versatile language fortwo Cdimensional shape operators and the description of (direct or inverse) wavefront re .ection/refraction problems in optics.It is also fundamental to the stability analysis of systems with uncertain parameters,in the context of the Routh CHurwitz criterion and the Kharitonov robustness theorem.