Current problems worked on include devising interpolation algorithms that use only piecewise ph curves, designing ph curves from geometric constraints, and designing with ph curves and surfaces. This research elegantly extends classical geometry work on cyclographic maps and Laguerre transforms that reformulate the problem of finding such curves and surfaces as an equivalent, simpler problem in a higher dimensional curved space. Research is coming close to finding practical interpolation and design algorithms for ph curves, but the corresponding problems for surfaces are largely open at this time. A property of practical interest is that the arc length of ph curves can be computed exactly, and this may impact NC path generation. These diverse subjects are related by the fact that they employ techniques from symbolic algebraic computation and algebraic geometry.
Geometric constraint solving is about solving systems of algebraic equations.
By judiciously restricting the problem and exploiting the geometric nature of the objects manipulated, very efficient algorithms can be obtained that have a strong impact on engineering design and analysis. Implicitization of parametric curves and surfaces is a test case for symbolic algebraic computations. Current research seeks to generalize some techniques originally developed at the beginning of this century. The technology is sophisticated, but its impact on practice appears to be limited. The geometry of implicit algebraic surfaces is investigated in the hope of finding a rich class of such surfaces that can be used in free-form design.