B-spline

CAD/CAM Subject-Mechanical engineering

1. B-Spline Curves

2. 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 curveof degree 3defined by 8
control points .
B-Spline Basis: Motivation

3. Those little dotssubdividethe B-spline curve into curve
segments.
One can move control pointsfor modifying the shape of the
curve just like what we do to Bézier curves.
We can also modifythe subdivision of the curve. Therefore,
B-spline curves have higher degree of freedomfor curve
design.
B-Spline Basis: Motivation

4. B-Spline Basis: Motivation
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
subdividedby points called knots.
These knots be 0 <= u
0<= u
1<= ... <= u
m <= 1.
Modifyingthe subdivision of [0,1] changes the shape of the
curve.

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

6. B-Spline Basis: Motivation
The computation of the coefficients is
perhaps the most complex step because they
must ensure certain continuity conditions.

7. 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)

8. 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)
Q
3is defined by P
0,P
1,P
2,P
3
Q
4is defined by P
1,P
2,P
3,P
4
Q
5is defined by P
2,P
3,P
4,P
5

Each curve segment
shares control points.

9. B-Spline Curves
(Two Advantages)
The effect of changingthe position of control
point P
4 (locality property).

10. B-Spline Basis Functions
(Knots, Knot Vector)
Let Ube a set of m+ 1non-decreasing
numbers, u
0 <= u
2 <= u
3<= ... <= u
m. The u
i's
are called knots,
The setUis the knot vector. 
muuuuU ,,,,
210 
u
1
u
0u
2
u
3
u
4u
5

11. B-Spline Basis Functions
(Knots, Knot Vector)
The half-open interval [u
i, u
i+1)is the i-th knot
span.
Some u
i's may be equal, some knot spansmay
not exist. 
muuuuU ,,,,
210 

12. B-Spline Basis Functions
(Knots)
If a knotu
iappearsk times (i.e., u
i= u
i+1 = ... =
u
i+k-1), where k > 1,u
iis a multiple knotof
multiplicity k, written asu
i(k).
If u
iappears only once, it is a simple knot.
If theknotsare equally spaced(i.e., u
i+1-u
iis a
constantfor 0 <= i <= m -1), the knot vector or
the knot sequence is saiduniform; otherwise, it
isnon-uniform. 
muuuuU ,,,,
210 

13. B-Spline Basis Functions
All B-spline basis functions are supposed
to have their domain on [u
0, u
m].
We use u
0= 0and u
m= 1frequently so that
the domainis the closed interval [0,1].

14. B-Spline Basis Functions
To define B-spline basis functions, we need one
more parameter.
The degreeof these basis functions,p. The i-th B-
spline basis function of degree p, written as
N
i,p(u), is defined recursively as follows:)()()(
otherwise0
if1
)0(
1,1
11
1
1,,
1
0,
uN
uu
uu
uN
uu
uu
uN
uuu
N
pi
ipi
pi
pi
ipi
i
pi
ii
i














 


15. B-Spline Basis Functions
To understand the way of computing N
i,p(u)for p
greater than 0, we use the triangular computation
scheme.

16. B-Spline Curves
(Definition)
Given n+ 1control points P
0, P
1, ..., P
nand a knot vector
U= { u
0, u
1, ..., u
m},the B-spline curve of degree pdefined
by these control pointsand knot vector Uis
The point on the curvethat corresponds to a knot u
i, C(u
i), is
referred to as a knot point.
The knot pointsdividea B-spline curve into curve segments,
each of which is defined on a knot span. 1,)()(
0
0
,


nmpuuuuNu
m
n
i
ipi
pC

17. B-Spline Curves
(Definition)
The degree of a B-spline basis function is an input.
To changetheshape 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 thecurve1,)()(
0
0
,


nmpuuuuNu
m
n
i
ipi
pC