Line Problems in Non-linear Computational Geometry

Frank Sottile
Texas A\&M University


  Geometric transversals, or lines which meet some or all of a
collection of objects, have long been studied in computational 
geometry.  That investigation considers geometric transversals 
from many points of view, including convexity, configuration,
combinatorics, and complexity.   My talk will concern the use 
of algebraic (both computational algebraic and real algebraic 
geometry) methods which have been recently applied to the 
solution of a particular problem of geometric transversals.

  This is an algebraic core problem for questions concerning 
transversals (or degenerate lines of sight) to polytopes and 
spheres.  The problem was to determine the number of common 
transversals to k lines that are also tangent to 4-k spheres
(or quadrics) when the lines and quadrics are general, and 
determine the possible degenerate configurations of the lines 
and quadrics.  While this investigation is very interesting 
both for algebraic geometry and for computational geometry, 
my focus will be to explain some ideas and techniques which 
may be applicable to a wider range of problems from computational 
geometry.