Bézier AwardTitle SMARida Farouki

Rida Farouki, the 2021 Pierre Bézier Award Recipient

Rida Farouki is famous for his introduction of Pythagorean hodograph (PH) curves in 1990 and his successive extensive research on these curves. Already prior to that, he had been a highly respected researcher in Computer-Aided Geometric Design (CAGD), especially known for his work on the numerical stability of the Bernstein form and on the interplay of algebraic and differential geometry in CAGD, in connection with offsets, surface-surfaces intersections and shape analysis. It is exactly this interplay where PH curves enter the scene. Polynomial PH curves are characterized by a polynomial arc length function. Note that an arc length parameterization is the preferred tool in Differential Geometry of curves, while rational parameterizations are studied in Algebraic Geometry. Linking these fields has been highly novel and led to many surprising and elegant results, from a purely mathematical perspective and from the perspective of applications in various manufacturing processes. The theoretical results include explicit representations of PH curves using complex numbers (for 2D curves) or quaternions (in 3D), complete classifications of the low degree cases (the most practical ones), approaches to design and approximation with these curves, to name just some of the many contributions. Some properties of PH curves laid the basis for entire series of papers by other researchers in the field, an example of that being the many studies on rational curves and surfaces with rational offsets. It would lead too far to just outline here the impact of this seminal work by Farouki. Convincing support for this claim is provided by Farouki’s book “Pythagorean-Hodograph Curves” which appeared as Volume 1 in the Springer Series “Geometry and Computing” (2008) and lists nearly 500 references related to the topic. In the meanwhile, the number of articles on or closely related to this subject may well have doubled.

The geometric and algebraic properties of PH curves turned out to be highly useful for those applications which Farouki had in mind from the very beginning, namely high-precision manufacturing, especially CNC machining. Not only the presence of exact offsets, but also the ease with which feedrates and other important machining parameters can be computed and optimized, make them ideal candidates for this field of applications. In more recent work, Farouki and his coworkers studied 3D PH curves which possess rational rotation minimizing frame motions and thus offer new possibilities in motion design and robotics. Farouki’s research in CNC machining is not restricted to relations to PH curves. In more recent work he addresses various important tasks (inverse kinematics, inverse dynamics problems) for high precision machining.

Apart from CAD/CAM, CNC machining and robotics, his interest in applications include rapid prototyping and geometrical optics, not to mention his early work in molecular dynamics and Monte Carlo simulation schemes, strongly–coupled plasmas, stellar dynamics, and the gravitational N-body problem. High precision is an important aspect in the work of Rida Farouki, not only in his applications in machining. He is always looking for precise, if possible, even explicit mathematical solutions. Therefore, he works not only with numerical methods, but is also heavily involved in symbolic computation and algebraic methods. Rida Farouki possesses an extraordinarily deep knowledge in many fields related to his area of research. He gives a great demonstration of this breadth in his book on Pythagorean Hodograph Curves or in a brilliant overview article on the Bernstein basis.

Rida Farouki belongs to the most highly cited researchers in his area, which is not surprising because his research, combining several areas of Mathematics (Differential Geometry, Algebraic Geometry, Approximation Theory) with various applications related to Manufacturing, is highly novel and inspired a large stream of follow-up work.

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