Computer Aided Engineering Design

132 COMPUTER AIDED ENGINEERING DESIGN

5.2 Why Splines?

The motivation is to develop Bernstein polynomials like basis functions Ψi(t) inheriting advantages
of barycentric (non-negativity and partition of unity) properties, with a difference that such properties
be local, that is, for parameter values of t∉ [t 0 ,t 1 ], it is desired that Ψi(t) = 0 while for t∈ [t 0 ,t 1 ],
Ψi(t) > 0. By intuition, we may expect Ψi(t) to be like a bell-shapedfunction shown in Figure 5.2.
Further, if Ψi(t) is an nthorder spline, a linear combination of such weights will inherently be
Cn–2continuous. Below are discussed various ways of computing the splines in an attempt to mould
them into basis functions with local control properties. The treatment and notation of B-spline basis
functions, to a large extent, follows from [27].

Figure 5.2 Schematic of the basis function as a spline curve ΨΨΨΨΨi(t)

t 0 t 1

Ψi(t)

5.3 Polynomial Splines

LetΦ(t) be a polynomial spline that has values yi at parameter values ti,i= 0, 1,... , n, with
ti–1 < ti < ti+1. Further, let Φ(t) be a cubic spline in each subinterval [ti–1,ti], with Φ(t) and its
derivatives,Φ′(t) and Φ′′(t) all continuous at the junction points (ti,yi). The ti,i= 0, 1,... , n are
termed as knotsand [ti–1,ti],i= 1,... , n as knot spans. If the knots are equally spaced (i.e., ti+ 1 −
tiis a constant for i= 0, 1,... , n–1), the knot vector or the knot sequence is said to be uniform;
otherwise, it is non-uniform.
One way to construct a polynomial spline is as follows. Let Φi(t) represent the spline in the ith
span,ti≤t≤ti+1. For the first span t 0 ≤t≤t 1 ,Φ 0 (t 0 ) and Φ 0 (t 1 ) are known as y 0 and y 1 , respectively.
To get a cubic spline, however, two more conditions are required for which let Φ 0 ′()t 0 and Φ 00 ′′()t

ti–1 ti ti+1

yi+1

yi

yi–1

Figure 5.3 Schematic of a polynomial spline

φi–1(t)

φi(t)