Paul de Casteljau
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Paul de Casteljau (born 1930 in Besançon, France.) is a French physicist and mathematician. In 1959, while working at Citroën, he developed an algorithm for evaluating calculations on a certain family of curves, which would later be formalized and popularized by engineer Pierre Bézier, and the curves called De Casteljau curve or Bézier curves. De Casteljau's algorithm is widely used, with some modifications, as it is the most robust and numerically stable method for evaluating polynomials. Other methods, such as Horner's method and forward differencing, are faster for calculating single points but are less robust. De Casteljau's algorithm is still very fast for subdividing a De Casteljau curve or Bézier curve into two curve segments at an arbitrary parametric location.
De Casteljau curves
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Paul de Casteljau received the 2012 Bézier Award from the Solid Modeling Association (SMA). The SMA's announcement highlights de Casteljau's eponymous algorithm:
- Paul de Castlejau's contributions are less widely known than should be the case because he was not able to publish them until equivalent ideas had been reinvented independently by others, sometimes in a rather different form but now recognisably related. Because he was not permitted to publish his early work, we now call polynomials with a Bernstein basis "Bézier polynomials", although Bézier himself did not use control points but their first difference vectors as the coefficients. We also call the multilinear polynomials "blossoming", following Lyle Ramshaw who in turn credited de Casteljau with the underlying "polar approach" to the mathematical theory of splines. We do call the algorithm for the stable evaluation of the Bernstein-Bézier form for polynomials "de Casteljau algorithm" although it is Carl de Boor's more general result applying it to B-splines which is now widely used in CAD/CAM systems.
The SMA also quotes Pierre Bézier on de Casteljau's contributions:
- There is no doubt that Citroën was the first company in France that paid attention to CAD, as early as 1958. Paul de Casteljau, a highly gifted mathematician, devised a system based on the use of Berstein polynomials. ... the system devised by de Casteljau was oriented towards translating already existing shapes into patches, defined in terms of numerical data. ... Due to Citroën's policy, the results obtained by de Casteljau were not published until 1974, and this excellent mathematician was deprived of part of the well deserved fame that his discoveries and inventions should have earned him.
- "SMA 2012 Bézier Award Announcement"
- Pierre Bézier, The first years of CAD/CAM and the UNISURF CAD System," pp 13-26 in Fundamental Developments of Computer- Aided Geometric Modeling, ed L. Piegl, 1993
- (French) Paul De Casteljau, Courbes à pôles, INPI, 1959 [clarification needed (the patent number would be welcome)]
- (French) Paul De Casteljau, Surfaces à pôles, INPI, 1963 [clarification needed (the patent number would be welcome)]
- (French) Mathématiques et CAO. Vol. 2 : Formes à pôles, Hermes, 1986
- (French) Les quaternions: Hermes, 1987
- (French) Le Lissage: Hermes, 1990
- POLoynomials, POLar Forms, and InterPOLation, September 1992, Mathematical methods in computer aided geometric design II, Academic Press Professional, Inc.
- Andreas Müller, "Neuere Gedanken des Monsieur Paul de Faget de Casteljau", 1995
- "de Casteljau's autobiography: My time at Citroën". Computer Aided Geometric Design. 16 (7): 583–586. August 1999. doi:10.1016/S0167-8396(99)00024-2.