Computer Aided Engineering Design

SPLINES 153

(C) Strong convex hull property: The B-spline curve, b(t) is contained in the convex hull defined
by the polyline [bj, bj+ 1 ,... , bj+p– 1 ] for t in [tj+p– 1 ,tj+p). This convex hull is the subset of the
parent hull [b 0 , b 1 ,... , bn]
Fort in the knot span [tj+p–1,tj+p),j+p– 1 = 0,... m–1,p basis functions, i.e.Np,j+p(t),Np,j+p+ 1 (t)

... , Np,j+2p– 1 (t) are non-zero from the property in Section 5.7(D). As Np,p+k(t) is the coefficient of bk,
onlyp control points, namely, bj,bj+ 1 ,... , bj+p– 1 have non-zero coefficients for t∈ [tj+p– 1 ,tj+p).
These coefficients also sum to one (property in Section 5.7(E)] making them barycentric in nature
like the Bernstein polynomials. Hence their weighted average, b(t) must lie in the convex hull defined
byp data points, bj,bj+ 1 ,... , bj+p – 1. The term strong implies that this convex hull is the subset of
the original convex hull of n+1 control points. As t crosses tj+p,Np,j+p(t) becomes zero while
Np,j+ 2 p(t) becomes non-zero. Consequently, b(t) for t∈ [tj+p,tj+p+1) lies in the new convex hull
defined by [bj+ 1 ,bj+2,... , bj+p] which again is the subset of the parent hull. The convex hull
property is elucidated in Figure 5.17 for an open B-spline curve in Figure 5.16 (a) for Example 5.7.
For 3 ≤ t < 4, b(t) lies in the convex hull of (0, 0), (0, 1), (2, 3) and (2.5, 6). For 4 ≤t < 5, the new
convex hull is defined by (0, 1), (2, 3), (2.5, 6) and (5, 2) and so on. For 6 ≤t < 7, the convex hull
is given by the last four data points in the set.

Figure 5.16 Unclamped and clamped B-spline curves: (a) unclamped spline, (b) spline clamped

at one end and (c) spline clamped at both ends

0246 8

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