Editing Spline (mathematics) (section)

==History==

According to Gerald Farin, B-splines were explored as early as the nineteenth century by [[Nikolai Lobachevsky]] at [[Kazan University]] in Russia.<ref>Farin, G. E. (2002). ''Curves and surfaces for CAGD: a practical guide''. Morgan Kaufmann. p. 119.</ref>

Before computers were used, numerical calculations were done by hand. Although piecewise-defined functions like the [[sign function]] or [[step function]] were used, polynomials were generally preferred because they were easier to work with. Through the advent of computers, splines have gained importance. They were first used as a replacement for polynomials in interpolation, then as a tool to construct smooth and flexible shapes in computer graphics.

It is commonly accepted that the first mathematical reference to splines is the 1946 paper by [[Isaac Jacob Schoenberg|Schoenberg]], 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 shipbuilding industries. In the foreword to (Bartels et al., 1987), [[Robin Forrest]] describes "[[lofting]]", a technique used in the British aircraft industry during [[World War II]] to construct templates for airplanes by passing thin wooden strips (called "[[flat spline|spline]]s") through points laid out on the floor of a large design loft, a technique borrowed from ship-hull design. For years the practice of ship design had employed models to design in the small. The successful design was then plotted on graph paper and the key points of the plot were re-plotted on larger graph paper to full size. The thin wooden strips provided an interpolation of the key points into smooth curves. The strips 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 1960s based on work by [[J. C. Ferguson]] at [[Boeing]] and (somewhat later) by [[Malcolm Sabin|M.A. Sabin]] at [[British Aircraft Corporation]].

The word "spline" was originally an [[East Anglian English|East Anglian]] dialect word.

The use of splines for modeling automobile bodies seems to have several independent beginnings. Credit is claimed on behalf of [[Paul de Casteljau|de Casteljau]] at [[Citroën]], [[Pierre Bézier]] at [[Renault]], and [[Garrett Birkhoff|Birkhoff]], [[Garabedian]], and [[Carl R. de Boor|de Boor]] at [[General Motors Corporation|General Motors]] (see Birkhoff and de Boor, 1965), all for work occurring in the very early 1960s or late 1950s. At least one of de Casteljau's papers was published, but not widely, in 1959. De Boor's work at [[General Motors Corporation|General Motors]] resulted in a number of papers being published in the early 1960s, including some of the fundamental work on [[B-spline]]s.

Work was also being done at Pratt & Whitney Aircraft, where two of the authors of (Ahlberg et al., 1967) — the first book-length treatment of splines — were employed, and the [[David Taylor Model Basin]], by Feodor Theilheimer. The work at [[General Motors Corporation|General Motors]] is detailed nicely in (Birkhoff, 1990) and (Young, 1997). Davis (1997) summarizes some of this material.