Exact Minkowski Products of N Complex Disks

A Disk-Covering Problem with Application in Optical Interferometry

Given a disk O in the plane called the objective, we want to find n small disks P1, . . . , Pn called the pupils such that ⋃n i,j=1 Pi ⊖ Pj ⊇ O, where ⊖ denotes the Minkowski difference operator, while minimizing the number of pupils, the sum of the radii or the total area of the pupils. This problem is motivated by the construction of very large telescopes from several smaller ones by so-calle...

Minkowski Geometric Algebra of Complex Sets

A geometric algebra of point sets in the complex plane is proposed, based on two fundamental operations: Minkowski sums and products. Although the (vector) Minkowski sum is widely known, the Minkowski product of two-dimensional sets (induced by the multiplication rule for complex numbers) has not previously attracted much attention. Many interesting applications, interpretations, and connection...

THE ROPER-SUFFRIDGE EXTENSION OPERATORS ON THE CLASS OF STRONG AND ALMOST SPIRALLIKE MAPPINGS OF TYPE $beta$ AND ORDER $alpha$

Let$mathbb{C}^n$ be the space of $n$ complex variables. Let$Omega_{n,p_2,ldots,p_n}$ be a complete Reinhardt on$mathbb{C}^n$. The Minkowski functional on complete Reinhardt$Omega_{n,p_2,ldots,p_n}$ is denoted by $rho(z)$. The concept ofspirallike mapping of type $beta$ and order $alpha$ is defined.So, the concept of the strong and almost spirallike mappings o...

Isometric Embedding of Negatively Curved Disks in the Minkowski Space

Computational Aspects of the Implementation of Disk Inversions

More than three decades the implementation of iterative methods for the simultaneous inclusion of polynomial zeros in circular complex interval arithmetic is carried out using the exact inversion of disks. Based on theoretical analysis and numerical examples, we show that the centered inversion gives smaller inclusion disks. This surprising result is the consequence of better convergence of the...