Spline Curves

Knots

Supports

Uniform vs. Non-Uniform

Basis functions

Partition of Unity

 


Spline Curves

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 spline.
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.