The idea of a spline function For many applications in modelling, the
interpolation and approximation methods discussed in the previous section
are not adequate. Usually the user does not want the curvature of the
curve or surface to vary too much, i.e., it should appear to be smooth.
Most of the classical interpolating functions (and in particular polynomials)
have a tendency to oscillate (see the figures). These figures show that
at several points, the interpolating polynomial deviates significantly
from the desired curve, suggesting that it may not be a good idea to
use polynomials to fit functions. One way to improve the situation is
to divide the given interval into smaller subintervals, and to construct
an approximating curve consisting of pieces of curves. This idea has
been used with considerable success for designing numerical quadrature
formulae. For example, the trapezoidal rule uses piecewise linear functions,
while Simpson's rule uses piecewise quadratic functions. In many applications,
however, we would like our curves to be smoother. Thus, it seems appropriate
to consider piecewise polynomial functions whose first derivatives (or
more generally whose first k derivatives) join continously.
Definition of Spline Curves A piecewise function s consisting of polynomial
pieces of degree n is called a) a spline function provided that it is
(n-1)-times continuously differentiable. b) a subspline function provided
that it is at least continuous, but it is not smooth enough to be a
Remark: The word spline comes from the name of a tool (consisting of
a thin flexible rod) used by ship builders for drawing smooth curves
approximating the cross sections of ship hulls.
Example of Subsplines Conic Sections as Subsplines: In engineering
applications (eg. in the aircraft industry) profiles are frequently
modelled by pieces of conic sections joined together to form a continuous
curve. The figure shows a typical such curve consisting of: Reflection
and translation of this profile produces a cylidrical object; if it
is rotated about the z-axis, we get a surface of revolution.