1990 Problem B: The Single Helix

A small biotechnological company must design, prove, program, and test a mathematical algorithm to locate “in real time” all the intersections of a helix and a plane in general positions in space.

Computer Aided Geometric Design (CAGD) programs enable engineers to view a plane section of an object they design, such as an automobile suspension or a medical device.  Engineers may also display on the plane section quantities such as air flow, stress, or temperature, coded by colors or level curves.  Plane sections may be rapidly swept through the entire object and its reactions to motion, forces, or heat.  To achieve such results, the computer programs must quickly and accurately locate all the intersections of the viewed plane and every part of the designed object.  General “equation solvers” may in principle compute such intersections, but for specific problems, specific methods may prove faster and more accurate than general methods.  In particular, general CAGD software may prove too slow to complete computations in real time, or too large to fit in the company’s finished medical devices.  These considerations have led the company to the following problem.

Design, justify, program, and test a method to compute all the intersections of a plane and a helix, both in general positions (at any locations and with any orientations) in space.  A segment of the helix may represent, for example, a helicoidal suspension spring or a piece of tubing in a chemical or medical apparatus.

Theoretical justification of the proposed algorithm is necessary to verify the solution from several points of view, for instance, through mathematical proofs of parts of the algorithm, and through test of the final program with known examples.  Such documentation and tests will be required by government agencies for medical use.