------------------------------------------------------- From: James Epperson Date: Thu, 9 Jul 1998 14:37:35 -0500 (CDT) Subject: History of Splines Two weeks ago I posted a query here about the connections between the development of spline approximations and automotive body design, and received over 30 responses, all of them very informative. Since several of them asked me to pass along what I learned, I decided to write up a brief summary and post it to the digest. Hence this note. It is commonly accepted that the first mathematical reference to splines is Schoenberg's paper [S], which is probably the first place that the word "spline" is used in connection with smooth, piecewise polynomial approximation. However, the ideas have their roots in the aircraft and ship-building industries. In the forward to [BBB], Robin Forrest describes "lofting," a technique used in the British aircraft industry during World War Two to construct templates for airplanes by passing thin wooden planks through points laid out on the floor of a large design loft. The planks would be held in place at discrete points (called "ducks" by Forrest; Schoenberg used "dogs" or "rats") and between these points would assume shapes of minimum strain energy. According to Forrest, one possible impetus for a mathematical model for this process was the potential loss of the critical design components for an entire aircraft should the loft be hit by an enemy bomb. This gave rise to "conic lofting," which used conic sections to model the position of the curve between the ducks. Conic lofting was replaced by what we would call splines in the early 1960's based on work by J. C. Ferguson at Boeing and (somewhat later) by M.A. Sabin at British Aircraft. Interestingly, Forrest says that the word "spline" comes from an East Anglian dialect. The use of splines for modeling automobile bodies seems to have several independent beginnings. Credit is claimed on behalf of de Casteljau at Citroen, Bezier at Renault, and Birkhoff, Garabedian, and de Boor at General Motors, all for work occuring in the very early 1960's or late 1950's. At least one of de Casteljau's papers was published, but not widely, in 1959. De Boor's work at GM resulted in a number of papers being published in the early 60's, including some of the fundamental work on B-splines. Work was also being done at Pratt & Whitney Aircraft, where two of the authors of [ANW] (the first book-length treatment of splines) were employed, and the David Taylor Model Basin, by Feodor Theilheimer. The work at GM is detailed nicely in the article [B] and the retrospective [Y]. I was also pointed to the article [BdB] by several people, but our library does not have that volume, so I have not been able to see it for myself. Paul Davis summarized some of this material in SIAM News in 1996; see [D]. Again, my thanks to all who sent me messages and suggestions. References: [ANW] Ahlberg, Nielson, and Walsh, The Theory of Splines and Their Applications, 1967. [B] Birkhoff, "Fluid dynamics, reactor computations, and surface representation," in A History of Scientific Computation (Steve Nash, editor), 1990. [BBB] Bartels, Beatty, and Barsky, An Introduction to Splines for Use in Computer Graphics and Geometric Modeling, 1987. [BdB] Birkhoff and de Boor, "Piecewise polynomial interpolation and approximation," Proc. General Motors Symposium of 1964, H. L. Garabedian, ed., Elsevier, New York and Amsterdam, 1965, pp. 164-190. [D] Davis, "B-splines and Geometric design," SIAM News, vol. 29, no. 5; available at http://www.wpi.edu/~pwdavis/sinews/spline17.htm. [S] Schoenberg, "Contributions to the problem of approximation of equidistant data by analytic functions," Quart. Appl. Math., vol. 4, pp. 45-99 and 112-141. [Y] Young, "Garrett Birkhoff and applied mathematics," Notices of the AMS, vol. 44, no. 11, pp. 1446-1449. ------------------------------ From: Peter Alfeld Date: Tue, 7 Jul 1998 07:53:50 -0600 (MDT) Subject: Bernstein-Bezier Applet For my teaching and research I have written an applet that lets you explore the Bernstein-Bezier Form of a bivariate polynomial, design finite elements, and analyze spaces of splines defined on triangulations. If you are interested check it out at http://www.math.utah.edu/~alfeld/MDS/ Peter Alfeld, Dept of Mathematics, University of Utah, Utah 84112, alfeld@math.utah.edu ------------------------------