Curves in Computer Graphics - Computer Graphics and Geometric Modeling - page 408

another example of this process shortly when we describe the uniform quadratic
B-splines.

11.5.1.4 Theorem. The functions N i,k (u) defined by equations (11.69) satisfy the

following properties:

(1) (Compact support property) The function N i,k (u) vanishes on (-•,u i ) » [u i+k ,•).
In particular, only the functions N i-k+1,k (u), N i-k+2,k (u),..., N i,k (u) are nonzero on
[u i ,u i+1 ).
(2) (Differentiability property) The function N i,k (u) is C • on the interior of the
spans and C k-1-m at a knot of multiplicity m. In particular, N i,k (u) is a spline of order
k if all knots have multiplicity 1.
(3) N i,k (u) ≥ 0, for all u.
(4) The identity

n

 0

, () =

Nu 1

ik

i =

holds for all u Π[k - 1,n + 1] and fails for any other u. If the knot vector is clamped,
then the identity holds for all u Π[0,n + k].

Proof. See [Seid89], [PieT95], or [Fari97]. Parts (1) and (3) are easy to prove using
induction on k. Note that they are trivially true when k is equal to 1. Induction also
plays a big role in the proof of the other parts. The reason for the restricted domain
in part (4) will become clearer in our discussion of uniform quadratic B-splines below.
Also, if the knot vector is clamped, then [k - 1,n + 1] = [0,n + k].

The compact support property of B-splines is important because it means that one
can make local modifications to curves based on such functions without having to
recompute the whole curve. Theorem 11.5.1.4(3) and (4) show that the B-spline basis
functions act like barycentric coordinates, which will be important for convexity
issues later.

Definition. Given a sequence of points p i , i = 0,1,...,n, the curve

n

Â

() = ()

pu N u p . (11.72)

ik , i

i = 0

is called the B-spline curve of order k (or degree m 1 ) with control or de Boor

points p i and knot vector (u 0 ,u 1 ,...,u n+k ). The adjectives clamped , unclamped , uniform ,
periodic , or nonuniform are applied to the curve if they apply to its knot vector. The
domain of the curve is defined to be the interval [u k-1 ,u n+1 ]. (Note that if the curve is
clamped, then the domain is the whole interval [u 0 ,u n+k ].) Each piece p([u i ,u i+1 ]), k-1
£ i £ n, of the whole curve traced out by p(u) is called a segment of the curve. The
polygonal curve defined by the control points is called the de Boor or control polygon
of the curve.

= k -

The reason that nonclamped B-spline curves have a restricted domain is that we
want the identity in Theorem 11.5.1.4(4) to hold.