Complex fans: a representation for vectors in polar form with interval attributes

ABSTRACT

If we allow the magnitude and angle of a complex number (expressed in polar form) to range over an interval, it describes a semicircular region, similar to a fan; these regions are what we call complex fans. Complex numbers are a special case of complex fans, where the magnitude and angle are point intervals. Operations (especially addition) with complex numbers in polar form are complicated. What most applications do is to convert them to rectangular form, perform operations, and return the result to polar form. However, if the complex number is a Complex Fan, that transformation increases ambiguity in the result. That is, the resulting Fan is not the smallest Fan that contains all possible results. The need for minimal results took us to develop algorithms to perform the basic arithmetic operations with complex fans, ensuring the result will always be the smallest possible complex fan. We have developed the arithmetic operations of addition, negation, subtraction, product, and division of complex fans. The algorithms presented in this article are written in pseudocode, and the programs in Common Lisp, making use of CLOS (Common Lisp Object System). Translation to any other high-level programming language should be straightforward.

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INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.4 MATHEMATICAL SOFTWARE
      Subjects: Algorithm design and analysis

Additional Classification:
  I. Computing Methodologies
  I.2 ARTIFICIAL INTELLIGENCE

  J. Computer Applications
  J.2 PHYSICAL SCIENCES AND ENGINEERING
      Subjects: Engineering

General Terms:
Algorithms

Keywords:
abstract data type, complex fans, complex numbers, interval computation, qualitative reasoning