Interpolating cubic splines

A spline is a thin flexible strip composed of a material such as bamboo or steel that can be bent to pass through or near given points in the plane, or in 3-space in a smooth manner. Mechanical engineers and drafting specialists find such (physical) splines useful in designing and in drawing plans for a wide variety of objects, such as for hulls of boats or for the bodies of automobiles where smooth curves need to be specified. These days, physi­ cal splines are largely replaced by computer software that can compute the desired curves (with appropriate encouragment). The same mathematical ideas used for computing "spline" curves can be extended to allow us to compute "spline" surfaces. The application ofthese mathematical ideas is rather widespread. Spline functions are central to computer graphics disciplines. Spline curves and surfaces are used in computer graphics renderings for both real and imagi­ nary objects. Computer-aided-design (CAD) systems depend on algorithms for computing spline functions, and splines are used in numerical analysis and statistics. Thus the construction of movies and computer games trav­ els side-by-side with the art of automobile design, sail construction, and architecture; and statisticians and applied mathematicians use splines as everyday computational tools, often divorced from graphic images.