B-spline

Consider designing the profile of a vase.
 The left figure below is a Bézier curve of degree 11; but, it is difficult to
bend the "neck" toward the line segment P4P5.
 The middle figure above uses this idea. It has three Bézier curve
segments of degree 3 with joining points marked with yellow rectangles.
 The right figure above is a B-spline curve of degree 3 defined by 8
control points .
B-Spline Basis: Motivation

 Those little dots subdivide the B-spline curve into curve
segments.
 One can move control points for modifying the shape of the
curve just like what we do to Bézier curves.
 We can also modify the subdivision of the curve. Therefore,
B-spline curves have higher degree of freedom for curve
design.

 Subdividing the curve directly is difficult to do. Instead, we
subdivide the domain of the curve.
 The domain of a curve is [0,1], this closed interval is
subdivided by points called knots.
 These knots be 0 <= u0 <= u1 <= ... <= um <= 1.
 Modifying the subdivision of [0,1] changes the shape of the
curve.

 In summary: to design a B-spline curve,
we need a set of control points, a set of
knots and a set of coefficients, one for each
control point, so that all curve segments are
joined together satisfying certain continuity
condition.

 The computation of the coefficients is
perhaps the most complex step because they
must ensure certain continuity conditions.

B-Spline Curves
(Two Advantages)
1. The degree of a B-spline polynmial can
be set independently of the number of
control points.
2. B-splines allow local control over the
shape of a spline curve (or surface)

B-Spline Curves
(Two Advantages)
 A B-spline curve that is defined by 6 control point,
and shows the effect of varying the degree of the
polynomials (2,3, and 4)
 Q3 is defined by P0,P1,P2,P3
 Each curve segment
shares control points.

B-Spline Curves
(Two Advantages)
 The effect of changing the position of control
point P4 (locality property).

B-Spline Basis Functions
(Knots, Knot Vector)
 Let U be a set of m + 1 non-decreasing
numbers, u0 <= u2 <= u3 <= ... <= um. The ui's
are called knots,
 The set U is the knot vector.
 
m
u
u
u
u
U ,
,
,
, 2
1
0


u1
u0 u2 u3 u4 u5

(Knots, Knot Vector)
 The half-open interval [ui, ui+1) is the i-th knot
span.
 Some ui's may be equal, some knot spans may
not exist.
 
m
u
u
u
u
U ,
,
,
, 2
1
0



(Knots)
 If a knot ui appears k times (i.e., ui = ui+1 = ... =
ui+k-1), where k > 1, ui is a multiple knot of
multiplicity k, written as ui(k).
 If ui appears only once, it is a simple knot.
 If the knots are equally spaced (i.e., ui+1 - ui is a
constant for 0 <= i <= m - 1), the knot vector or
the knot sequence is said uniform; otherwise, it
is non-uniform.
 
m
u
u
u
u
U ,
,
,
, 2
1
0



All B-spline basis functions are supposed
to have their domain on [u0, um].
 We use u0 = 0 and um = 1 frequently so that
the domain is the closed interval [0,1].

 To define B-spline basis functions, we need one
more parameter.
 The degree of these basis functions, p. The i-th B-
spline basis function of degree p, written as
Ni,p(u), is defined recursively as follows:
)
(
)
(
)
(
otherwise
0
if
1
)
0
(
1
,
1
1
1
1
1
,
,
1
0
,
u
N
u
u
u
u
u
N
u
u
u
u
u
N
u
u
u
N
p
i
i
p
i
p
i
p
i
i
p
i
i
p
i
i
i
i


















 



 To understand the way of computing Ni,p(u) for p
greater than 0, we use the triangular computation
scheme.

B-Spline Curves
(Definition)
 Given n + 1 control points P0, P1, ..., Pn and a knot vector
U = { u0, u1, ..., um }, the B-spline curve of degree p defined
by these control points and knot vector U is
 The point on the curve that corresponds to a knot ui, C(ui), is
referred to as a knot point.
 The knot points divide a B-spline curve into curve segments,
each of which is defined on a knot span.
1
,
)
(
)
( 0
0
, 




 

n
m
p
u
u
u
u
N
u m
n
i
i
p
i p
C

B-Spline Curves
(Definition)
 The degree of a B-spline basis function is an input.
 To change the shape of a B-spline curve, one can modify
one or more of these control parameters:
1. The positions of control points
2. The positions of knots
3. The degree of the curve
1
,
)
(
)
( 0
0
, 




 

n
m
p
u
u
u
u
N
u m
n
i
i
p
i p
C

B-spline

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B-spline