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.
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.
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