Numerical modeling of the conformational transition of a spiral focusing surface

ABSTRACT

A new class of focusing reflectors was created by modeling the conformational transitions between a wound spiral focusing surface and its planar, non-focusing, unwound spiral state. Two solutions to the problem were found; one by modeling the process of unwinding the focusing spiral surface, the other by modeling the winding of the planar spiral surface. Both the mathematical derivations and the resulting equations relating the two states of the spiral surface proved markedly different for the two approaches. Implementation of each solution was accomplished by specifying the locations of marker points and tracking them through the conformational transitions. Although the mathematics and the algorithms employed are completely different between the two solutions, both result in identical sets of marker points for a given spiral. However, the solution based on the winding conformational transition is completely general, while the solution based on unwinding is limited to a narrow set of focusing geometries.

REFERENCES

• 1: Steenblik, R.A. and D.V. Ho, 1982. '~resnel Spiral Reflector and Method for Making Same." United States Patent number 4,350,412.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS

Additional Classification:
  D. Software
  D.3 PROGRAMMING LANGUAGES
      D.3.2 Language Classifications

          Nouns: FORTRAN

  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems
          Subjects: Geometrical problems and computations

General Terms:
Algorithms, Languages, Theory