\chapter{A Brief History of Splines} The following text is blatantly ripped off from James Epperson: 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]. End rip-off. Schoenberg \cite{Sch46} was one of the first to propose cubic splines as a small-angle approximation to elastica. [Note: what follows is the corrected version as appears in his selected papers\cite{SchoenbergPapers}] \textbf{3.1 Polynomial spline curves of order $k$.} A spline is a simple mechanical device for drawing smooth curves. It is a slender flexible bar made of wood or some other elastic material. The spline is placed on the sheet of graph paper and held in place at various points by means of certain heavy objects (called ``dogs'' or ``rats'') such as to take the shape of the curve we wish to draw. Let us assume that the spline is so placed and supported as to take the shape of a curve which is nearly parallel to the $x$-axis. If we denote by $y = y(x)$ the equation of this curve then we may neglect its small slope $y'$, whereby its curvature becomes \[ 1/R = y'' / (1 + y'^2)^{3/2} \approx y'' \] The elementary theory of the beam will then show that the curve $y = y(x)$ is a polygonal line composed of cubic arcs which join continuously, with a continuous first and second derivative\footnote{Schoenberg is indebted for this suggestion to Professor L. H. Thomas of Ohio State University}. These junction points are precisely the points where the heavy supporting objects are placed. 3.11. \textit{Description of spline curves of order $k$.} Our last remark suggests the following definition. {\sc Definition} 4. \textit{A real function $F(x)$ defined for all real $x$ is called a spline curve of order $k$ and denoted by $\Pi_k(x)$ if it enjoys the following properties:} 1) \textit{It is compossed of polynomial arcs of degree at most $k - 1$.} 2) \textit{It is of class $C^{k - 2}$, i.e., $F(x)$ has $k=2$ continuous derivatives.} 3) \textit{The only possible junction points of the various polynomial arcs are the integer points $x=n$ if $k$ is even, or else the points $x = n + 1/2$ if $k$ is odd.}