Spline

Etymology

The term spline comes from a mechanical device traditionally used to draw curves, composed of long strips of a flexible but elastic material (usually wood, metal, or plastic). After being developed in the 1940s, spline functions were quickly adopted in computeraided design software to replace the mechanical technique, by which they eventually found their way into GIS. As an interpolation technique, splines came to GISci through statistics.

Mathematics

In its simplest, univariate form, a spline S divides the domain D = [a, b] of a curve into a number of pieces Di = [ti-1, ti], such that i∊[1, n], t0 = a and tn = b. Each ti is called a knot or node; in most applications, the function value at each knot si = S(ti) is given. A distinct polynomial function fi:Di → R is defined over each piece, such that the functions are continuous at each knot

but they are not necessarily differentiable at each knot. Thus, for a cubic spline, each function is defined as

The parameters ai, bi, ci, and di can be determined by several methods, two of which are most common. A natural spline is the smoothest possible curve that passes through every knot <ti, si > and has continuous first and second derivatives at each knot

and best approximating the original mechanical device. A Bézier curve does not pass through every knot; rather, of every four knots, it passes through the first and last, while the second and third are used to define a smooth polynomial. When drawing Bézier curves in graphics, computer-aided drafting (CAD), and GIS, the second and third knots are often shown as “handles” that appear to have a gravitational effect on the curve.

To represent curves in two-dimensional space (e.g., in a vector data model), the domain is generally a surrogate variable t, upon which two spline functions define the x-and y-coordinates of the curve. A 2.5D surface (e.g., terrain) can be defined by splines using a two-dimensional domain divided into twodimensional pieces (generally quadrilaterals with knots defined at each corner). Splines representing truly three-dimensional surfaces (with three input variables and three output variables) are commonly used in CAD, but rarely in GIS.