Graphics Reference
In-Depth Information
Definition.
Given n ≥ 0, k ≥ 1, and a nondecreasing sequence of real numbers U =
(u
0
,u
1
,...,u
n+k
), define functions
N
:
R
Æ£
,
0
i
£
n
,
ik
,
recursively as follows:
()
=
N
u
1
0
,
for u
£
u
<
u
,
and
i
,
1
i
i
+
1
=
,
elsewhere
.
(11.69a)
If k > 1, then
uu
-
Nu
uu
-
i
ik
ik
+
()
=
()
+
()
Nu
uu
Nu
,
(11.69b)
ik
,
ik
,
-
1
i
+-
11
,
k
u
-
u
-
ik
+-
1
i
+
i
+
1
where, if any term is of the form 0/0, we replace it by 0. The function N
i,k
(u) is called
the ith
B-spline
, or
B-spline basis function
,
of order k and degree k
-
1 with respect to the
knot vector U
.
Note.
We need to make one technical point. Although a B-spline of order k is a spline
of some order, it is not necessarily a spline of order k. If some knot has multiplicity
larger than 1, then a B-spline is not differentiable enough at that point. See Theorem
11.5.1.4(2). This causes an occasional awkwardness when talking about such func-
tions. On the other hand, it turns out that
any
spline is a linear combination of
B-splines basis functions (Theorem 11.5.2.16), so that one not lose anything if one
concentrates on these particular splines. The fact that B-splines form a basis for the
space of splines is actually what gave B-splines their name because the “B” in the
name stands for “basis” ([Scho67]).
A spline is greatly influenced by how the knots are chosen. Before we work
through some examples showing the shape of a few of the splines N
i,k
(u) it is con-
venient to introduce some terminology.
Definition.
A knot vector for a spline or a B-spline of order k is said to be
clamped
if the first and last knot each has multiplicity k. Otherwise, it is said to be
unclamped
.
An (unclamped) knot vector U of length L is said to be
uniform
or
periodic
if the knots
u
i
are evenly spaced, that is, there is a constant d > 0, so that u
i+1
= u
i
+d for 0 £ i £
L - 2. If U is clamped, then it is said to be
uniform
if all the knots u
i
except the first
and last k knots are evenly spaced, that is, u
i+1
= u
i
+d, for k £ i < L - k. A knot vector
that is not uniform is said to be
nonuniform
.
Definition.
The adjectives
clamped
,
unclamped
,
uniform
,
periodic
, or
nonuniform
are
applied to a spline or B-spline if they apply to its knot vector.
Note.
Unfortunately, the terminology for splines and their knot vectors did not
develop in a consistent way. Other terms can also be found in the literature. For
example, the term “open uniform” is sometimes used for what we are calling clamped
