B spline

On-Line Geometric Modeling Notes
DEFINITION OF A B-SPLINE CURVE
Kenneth I. Joy
Visualization and Graphics Research Group
Department of Computer Science
University of California, Davis
Overview
These notes present the direct definition of the B-Spline curve. This definition is given in two ways: first
by an analytical definition using the normalized B-spline blending functions, and then through a geometric
definition.
The B-Spline Curve – Analytical Definition
A B-spline curve P(t), is defined by
P(t) =
n
i=0
PiNi,k(t)
where
• the {Pi : i = 0, 1, ..., n} are the control points,
• k is the order of the polynomial segments of the B-spline curve. Order k means that the curve is made
up of piecewise polynomial segments of degree k − 1,
• the Ni,k(t) are the “normalized B-spline blending functions”. They are described by the order k and
by a non-decreasing sequence of real numbers
{ti : i = 0, ..., n + k} .

normally called the “knot sequence”. The Ni,k functions are described as follows
Ni,1(t) =



1 if u ∈ [ti, ti+1),
0 otherwise.
(1)
and if k > 1,
Ni,k(t) =
t − ti
ti+k−1 − ti
Ni,k−1(t) +
ti+k − t
ti+k − ti+1
Ni+1,k−1(t) (2)
• and t ∈ [tk−1, tn+1).
We note that if, in equation (2), either of the N terms on the right hand side of the equation are zero,
or the subscripts are out of the range of the summation limits, then the associated fraction is not evaluated
and the term becomes zero. This is to avoid a zero-over-zero evaluation problem. We also direct the readers
attention to the “closed-open” interval in the equation (1).
The order k is independent of the number of control points (n + 1). In the B-Spline curve, unlike
the Bézier Curve, we have the flexibility of using many control points, and restricting the degree of the
polymonial segments.
The B-Spline Curve – Geometric Definition
Given a set of Control Points {P0, P1, ..., Pn}, an order k, and a set of knots {t0, t1, ..., tn+k}, the
B-Spline curve of order k is defined to be
P(t) = P
(k−1)
l (t) if u ∈ [tl, tl+1)
where
P
(j)
i (t) =



(1 − τj
i )P
(j−1)
i−1 (t) + τj
i P
(j−1)
i (t) if j > 0,
Pi if j = 0.
and
τj
i =
t − ti
ti+k−j − ti
2

It is useful to view the geometric construction as the following pyramid
...
Pl−k+1
P
(1)
l−k+2
Pl−k+2 P
(2)
l−k+3
P
(1)
l−k+3 ·
Pl−k+3 ·
Pl−k+4
· P
(k−2)
l−1
· P
(k−1)
l
· P
(k−2)
l
·
Pl−2 P
(2)
l−1
P
(1)
l−1 ·
Pl−1 P
(2)
l
P
(1)
l
Pl
...
Any P in this pyramid is calculated as a convex combination of the two P functions immediately to it’s left.
All contents copyright (c) 1996, 1997, 1998, 1999
Computer Science Department, University of California, Davis
All rights reserved.
3

THE UNIFORM B-SPLINE BLENDING FUNCTIONS
Kenneth I. Joy
Overview
The normalized B-spline blending functions are deﬁned recursively by
Ni,1(t) =



1 if u ∈ [ti, ti+1)
0 otherwise
(1)
and if k > 1,
Ni,k(t) =
t − ti
ti+k−1 − ti
Ni,k−1(t) +
ti+k − t
ti+k − ti+1
Ni+1,k−1(t) (2)
where {t0, t1, ..., tn+k} is a non-decreasing sequence of knots, and k is the order of the curve.
These functions are difﬁcult to calculate directly for a general knot sequence. However, if the knot
sequence is uniform, it is quite straightforward to calculate these functions – and they have some suprising
properties.
Calculating the Blending Functions using a Uniform Knot Sequence
Assume that {t0, t1, t2, ..., tn} is a uniform knot sequence, i.e., {0, 1, 2, ..., n}. This will simplify the
calculation of the blending functions, as ti = i.
Blending Functions for k = 1

if k = 1, then by using equation (1), we can write the normalized blending functions as
Ni,1(t) =



1 if u ∈ [i, i + 1)
0 otherwise
(3)
These are shown together in the following ﬁgure, where we have plotted N0,1, N1,1, N2,1, and N3,1 respec-
tively, over ﬁve of the knots. Note the white circle at the end of the line where the functions value is 1. This
represents the affect of the “open-closed” interval found in equation (1).
2

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These functions have support (the region where the curve is nonzero) in an interval, with Ni,1 having support
on [i, i + 1). They are also clearly shifted versions of each other – e.g., Ni+1,1 is just Ni,1 shifted one unit
to the right. In fact, we can write Ni,1(t) = N0,1(t − i)
3

If k = 2 then N0,2 can be written as a weighted sum of N0,1 and N1,1 by equation (2). This gives
N0,2(t) =
t − t0
t1 − t0
N0,1(t) +
t2 − t
t2 − t1
N1,1(t)
= tN0,1(t) + (2 − t)N1,1(t)
=



t if 0 ≤ t 1
2 − t if 1 ≤ t 2
0 otherwise
This curve is shown in the following ﬁgure. The curve is piecewise linear, with support in the interval [0, 2].
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These functions are commonly referred to as “hat” functions and are used as blending functions in many
linear interpolation problems.
Similarly, we can calculate N1,2 to be
N1,2(t) =
t − t1
t2 − t1
N1,1(t) +
t3 − t
t3 − t2
N2,1(t)
= (t − 1)N1,1(t) + (3 − t)N2,1(t)
=



t − 1 if 1 ≤ t 2
3 − t if 2 ≤ t 3
0 otherwise
This curve is shown in the following ﬁgure, and it is easily seen to be a shifted version of N0,2.
4

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Finally, we have that
N1,2(t) =



t − 2 if 2 ≤ t 3
4 − t if 3 ≤ t 4
0 otherwise
which is shown in the following illustration.
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These nonzero portion of these curves each cover the intervals spanned by three knots – e.g., N1,2 spans the
interval [1, 3]. The curves are piecewise linear, made up of two linear segments joined continuously.
Sinve the curves are shifted versions of each other, we can write
Ni,2(t) = N0,2(t − i)
5

For the case k = 3, we again use equation (2) to obtain
N0,3(t) =
t − t0
t2 − t0
N0,2(t) +
t3 − t
t3 − t1
N1,2(t)
=
t
2
N0,2(t) +
3 − t
2
N1,2(t)
=



t2
2 if 0 ≤ t 1
t2
2 (2 − t) + 3−t
2 (t − 1) if 1 ≤ t 2
(3−t)2
2 if 2 ≤ t 3
0 otherwise
=



t2
2 if 0 ≤ t 1
−3+6t−2t2
2 if 1 ≤ t 2
(3−t)2
2 if 2 ≤ t 3
0 otherwise
and by nearly identical calculations,
N1,3(t) =
t − t1
t3 − t1
N1,2(t) +
t4 − t
t4 − t2
N2,2(t)
=
t − 1
2
N1,2(t) +
4 − t
2
N2,2(t)
=



(t−1)2
2 if 1 ≤ t 2
−11+10t−2t2
2 if 2 ≤ t 3
(4−t)2
2 if 3 ≤ t 4
0 otherwise
These curves are shown in the following ﬁgure. They are clearly piecewise quadratic curves, each made up
of three parabolic segments that are joined at the knot values (we have placed tick marks on the curves to
show where they join).
6

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The nonzero portion of these two curves each span the interval between four consecutive knots – e.g., the
nonzero portion of N1,3 spans the interval [1, 4]. Again, N1,3 can be seen visually to be a shifted version of
N0,3. (This fact can also be seen analytically by substituting t + 1 for t in the equation for N1,3.) We can
write
Ni,3(t) = N0,3(t − i)
Blending Functions of Higher Orders
It is not too difﬁcult to conclude that the Ni,4 blending functions will be piecewise cubic functions. The
support of Ni,4 will be the interval [i, i + 4] and each of the blending functions will be shifted versions of
each other, allowing us to write
Ni,4(t) = N0,4(t − i)
In general, the uniform blending functions Ni,k will be piecewise k − 1st degree functions having support
in the interval [i, i + k). They will be shifted versions of each other and each can be written in terms of a
“basic” function
Ni,k(t) = N0,k(t − i)
7

Summary
In the case of the uniform knot sequence, the blending functions are fairly easy to calculate, are shifted
versions of each other, and have support over a simple interval determined by the knots. These characteristics
are unique to the uniform blending functions.
For other special characteristics see the notes that describe writing the uniform blending functions as a
convolution, and the notes that describe the two-scale relation for uniform B-splines
All contents copyright (c) 1996, 1997, 1998, 1999, 2000
8

THE DEBOOR-COX CALCULATION
Kenneth I. Joy
Overview
In 1972, Carl DeBoor and M.G. Cox independently discovered the relationship between the analytic
and geometric definitions of B-splines. Starting with the definition of the normalized B-spline blending
functions, these two researchers were able to develop the geometric definition of the B-spline. It is this
calculation that is discussed in this paper.
Definition of the B-Spline Curve
A B-spline curve P(t), is defined by
P(t) =
n
i=0
PiNi,k(t) (1)
where
• the {Pi : i = 0, 1, ..., n} are the control points,
• k is the order of the polynomial segments of the B-spline curve. Order k means that the curve is made
up of piecewise polynomial segments of degree k − 1,
• the Ni,k(t) are the “normalized B-spline blending functions”. They are described by the order k and
by a non-decreasing sequence of real numbers
{ti : i = 0, ..., n + k} .

normally called the “knot sequence”. The Ni,k functions are described as follows
Ni,1(t) =



1 if t ∈ [ti, ti+1),
0 otherwise.
(2)
and if k 1,
Ni,k(t) =
t − ti
ti+k−1 − ti
Ni,k−1(t) +
ti+k − t
ti+k − ti+1
Ni+1,k−1(t) (3)
• the parameter t ranges throughout the interval [tk−1, tn+1).
We note that if, in equation (3), either of the N terms on the right hand side of the equation are zero, or
the subscripts are out of the range of the summation limits, then the associated fraction is not evaluated and
the term becomes zero. This is to avoid a zero-over-zero evaluation problem.
The order k is independent of the number of control points (n + 1). In the B-Spline curve, unlike
the Bézier Curve, we have the flexibility of using many control points, and restricting the degree of the
polymonial segments.
The DeBoor-Cox Calculation
In the DeBoor-Cox calculation, we substitute the definition of Ni,k(t) given in equation (3), into the
right-hand side of
P(t) =
n
i=0
PiNi,k(t)
and simplify. This will give us the definition of P(t) in terms of Ni,k−1, which is of lower degree. The
general idea is to continue this process until the sum is written with Ni,1 functions, which we can evaluate
easily.
So here we go. If t ∈ [tk−1, tn+1), then by substituting equation (3) into the equation (1), we have
2

P(t) =
n
i=0
PiNi,k(t)
=
n
i=0
Pi
t − ti
ti+k−1 − ti
Ni,k−1(t) +
ti+k − t
ti+k − ti+1
Ni+1,k−1(t)
Distributing the sums, we obtain
=
n
i=0
Pi
t − ti
ti+k−1 − ti
Ni,k−1(t) +
n
i=0
Pi
ti+k − t
ti+k − ti+1
Ni+1,k−1(t)
We now separate out those unique terms of each sum, N0,k−1 and Nn+1,k−1, giving
= P0
t − t0
tk−1 − t0
N0,k−1(t) +
n
i=1
Pi
t − ti
ti+k−1 − ti
Ni,k−1(t)
+
n−1
i=0
Pi
ti+k − t
ti+k − ti+1
Ni+1,k−1(t) + Pn
tn+k − t
tn+k − tn+1
Nn+1,k−1(t)
Now since the support of a B-spline blending function Ni,k(t) is the interval [ti, ti+k], we have that
N0,k−1 is non-zero only if t ∈ [t0, tk−1), which is outside the interval [tk−1, tn+1) (where P(t) is deﬁned).
Thus, N0,k−1(t) ≡ 0. Also Nn+1,k−1 is non-zero only if t ∈ [tn+1, tn+k−1), which is outside the interval
3

[tk−1, tn+1). Thus, Nn+1,k−1(t) ≡ 0 and we have
P(t) =
n
i=1
Pi
t − ti
ti+k−1 − ti
Ni,k−1(t) +
n−1
i=0
Pi
ti+k − t
ti+k − ti+1
Ni+1,k−1(t)
If we change the summation limits, we get
=
n
i=1
Pi
t − ti
ti+k−1 − ti
Ni,k−1(t) +
n
i=1
Pi−1
ti+k−1 − t
ti+k−1 − ti
Ni,k−1(t)
Combining the two sums, and rewriting, we obtain
=
n
i=1
ti+k−1 − t
ti+k−1 − ti
Pi−1 +
t − ti
ti+k−1 − ti
Pi Ni,k−1(t)
If we denote
P
(1)
i (t) =
ti+k−1 − t
ti+k−1 − ti
Pi−1 +
t − ti
ti+k−1 − ti
Pi
then the above result may be written
P(t) =
n
i=1
P
(1)
i (t)Ni,k−1(t)
Now we have written the summation terms of equation (1) in terms of blending functions of lower degree.
Of course, we have transferred some of the complexity to the P
(1)
i s, but we retain a similar form with control
points P
(1)
i s weighted by blending functions, and we can repeat this calculation again.
Once again then, repeating the calculation and manipulating the sums, we obtain
P(t) =
n
i=2
P
(2)
i (t)Ni,k−2(t)
where
P
(2)
i (t) =
ti+k−2 − t
ti+k−2 − ti
P
(1)
i−1(t) +
t − ti
ti+k−2 − ti
P
(1)
i (t)
4

(Note the appearance of the 2s on the right-hand side of the equation.)
If we continue with this process again, we will manipulate the sum so that the blending functions have
order k − 3. Then again with give us k − 4, and eventually we will obtain blending functions of order 1. We
are lead to the following result: If we define
P
(j)
i (t) =



(1 − τj
i )P
(j−1)
i−1 (t) + τj
i P
(j−1)
i (t) if j 0
Pi if j = 0
(4)
where
τj
i =
t − ti
ti+k−j − ti
Then, if t is in the interval [tl, tl+1), we have
P(t) = P
(k−1)
l (t)
This can be shown by continuing the DeBoor-Cox calculation k − 1 times. When complete, we arrive
at the formula
P(t) =
n
i=k−1
P
(k−1)
i (t)Ni,1(t)
where P
(j)
i (t) is given in equation (4). (Note the algebraic simplification that the τ’s provide.) If t ∈
[tl, tl+1], then then the only nonzero term of the sum is the lth term, which is one, and the sum must equal
P
(k−1)
l (t).
This enables us to define the geometrical definition of the B-spline curve.
Geometric Definition of the B-Spline Curve
Given a set of Control Points
P0, P1, ..., Pn
5

and a set of knots
t0, t1, ..., tn+k
The B-Spline curve of order k is deﬁned to be
P(t) = P
(k−1)
l (t)if t ∈ [tl, tl+1)
where
P
(j)
i (t) =



(1 − τj
i )P
(j−1)
i−1 (t) + τj
i P
(j−1)
i (t) if j 0
Pi if j = 0
and
τj
i =
t − ti
ti+k−j − ti
6

It is useful to view the geometric construction as the following pyramid
...
Pl−k+1
P
(1)
l−k+2
Pl−k+2 P
(2)
l−k+3
P
(1)
l−k+3 ·
Pl−k+3 ·
Pl−k+4
· P
(k−2)
l−1
· P
(k−1)
l
· P
(k−2)
l
·
Pl−2 P
(2)
l−1
P
(1)
l−1 ·
Pl−1 P
(2)
l
P
(1)
l
Pl
...
Any P
(j)
i in this pyramid is calculated as a convex combination of the two P functions immediately to it’s
left.
Summary
The DeBoor-Cox calculation is a fundamental result in the geometric modeling field. It was used to
exhibit the relation between the analytic definition of the B-Spline curve and the geometric definition of the
curve. The geometric definition of the curve, because of its computational stability has become the primary
technique by which points on these curves are calculated.
7

8

THE SUPPORT OF A
NORMALIZED B-SPLINE BLENDING FUNCTION
Kenneth I. Joy
Overview
A B-spline blending function Ni,k(t) has compact support. This means that the function is zero outside
of some interval. In these notes, We ﬁnd this interval explicitly in terms of the knot sequence.
The Support of the Function
Given an order k, and a knot sequence {t0, t1, t2, ..., tn+k}, the normalized B-spline blending function
Ni,k(t) is positive if and only if t ∈ [ti, ti+k).

We can show that this is true by considering the following pyramid structure.
Ni,1
. Ni+1,1
. .
. .
Ni,k−2 .
Ni,k−1
Ni,k Ni+1,k−2
Ni+1,k−1
Ni+2,k−2 .
. .
. .
. Ni+k−2,1
Ni+k−1,1
The deﬁnition of the normalized blending function Ni,k as a weighted sum of Ni,k−1(t) and Ni+1,k−1(t).
Thus for any of the N functions in the pyramid, it is a weighted sum of the two items immediately to its
right. If we follow the pyramid to its right edge, we see that the only blending functions Nj,1 that contribute
to Ni,k are those with i ≤ j ≤ i + k − 1, and these function are collectively nonzero when t ∈ [ti, ti+k).
Summary
A B-spline blending function has compact support. The support of this function depends on the knot
sequence and always covers an interval of containing several knots – containing k + 1 knots if the curve is
or order k.
2

B spline

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