CAD has many different aspects, ranging from the design of graphical user interfaces to algebraic and numerical problems in geometry. The research activity in the Department is concentrated on the latter, and especially on issues of computational robustness of the representation and manipulation of curves and surfaces.

Projects include

- High integrity data exchange between different CAD systems
- Numerically robust representations of curves and surfaces
- Resultants

High integrity data exchange between different CAD systems
is an important industrial requirement because it allows the smooth transfer
of CAD models between these diverse systems. Unfortunately this transfer
is often difficult, and in certain situations it may be impossible to reproduce
the CAD model *exactly* in the receiving system. The reasons
for this are mathematical - in some cases, data tarnsfer requires the solution
of an ill-conditioned (unstable) set of equations, and in other cases,
a solution does not exist, and approximations must therefore be made. The
research in the Department is focussed on addressing these issues of instability
and approximation.

It is recognised that the representation of curves and surfaces must be numerically stable, such that errors in the data do not propagate and invalidate the answers to computations that are performed on their polynomial coefficients. It is therefore necessary to quantify the numerical condition of a polynomial and algebraic curves and surfaces. New and interesting results have been obtained, and further work is aimed at considering the numerical condition of geometric singularities.

The determination of the points of intersection of two
curves is a problem that arises frequently in CAD systems. For example,
the intersection points of two implicitly defined curves are defined by
the common solutions of their polynomial forms. These solutions can be
found by the method of resultants, a technique that is well established
in algebraic geometry. Although resultants have a rich history and are
well understood theoretically, their computational implementation is non-trivial
because it is required to determine *integer* information (the rank
of a matrix) from *floating-point* data (polynomial coefficients).
This implies that an exact answer cannot be given, and that a tolerance
on the computed answer must be given. One of the research interests in
the Department is the numerically stable determination of resultants and
their application to CAD systems.

Papers and reports on these topics are in my publications page.

Click here for my home page.His e-mail address is j.winkler@dcs.shef.ac.uk