Graphics Reference
In-Depth Information
location of the control points, provides control over the shape of the curve
to be drawn. In B-spline, we have three different choices for knot values: the
uniform non-periodic B-spline knots, uniform periodic knots, and nonuniform
knots. Open curves are modeled by uniform non-periodic knots, while closed
curves are modeled by uniform periodic knots. Similarly, nonuniform knots
can also be of two different types: nonuniform non-periodic and nonuniform
periodic to model respectively open and closed curves.
Uniform Non-Periodic Knot Structure
The
m
th order or (m-1 degree) B-spline
B
i,m
(
u
),
i
=0
,
1
,
···
,n
is defined
for the parameter
u
m
+ 2].
B
i,m
represents a curve, known as the
B-spline curve. When the curve is a uniform open curve (non-periodic), its
uniform non-periodic knots
t
0
to
t
n
+
m
are chosen according to the following
rule:
∈
[0
,n
−
t
i
=0
if
0
≤
i<m
=
i
−
m
+1
if
m
≤
i
≤
n
(5.5)
=
n
−
m
+2
if
n < i
≤
n
+
m.
Example: Find the uniform non-periodic knot vector for a b-spline open curve
for which
n
=5and
m
=3.
We can note that knots range from
t
0
to
t
n
+
m
=
t
8
. According to equation
(5.5),
t
0
=
t
1
=
t
2
=0and
t
6
=
t
7
=
t
8
= 4. Besides, we have
t
3
=1,
t
4
=2,
and
t
5
= 3. The knot vector is, therefore, [0
,
0
,
0
,
1
,
2
,
3
,
4
,
4
,
4].
In general, the choice of knots according to the equation (5.5) is found to
provide the following knot structure for uniform non-periodic open curves,
0
,
0
,
0
m
knots
···
,
1
,
2
,
···
,n
−
m
+1
,n
−
m
+2
,n
−
m
+2
,
···
,n
−
m
+2
.
m
knots
The use of repeated knots ensures that the end points of the spline coincide
with the end points of the control polygon. Note that in the beginning, we
have
m
knots, at the end we have
m
knots, and in between we have
n
m
+1
knots. Therefore, the total number of knots for any open control polygon is
m
+(
n
−
−
m
+1)+
m
or
n
+
m
+1.
Uniform Periodic Knot Structure
When the B-spline curve is closed (periodic) and the spacing between the knot
values is fixed, the resulting curve is called a uniform periodic B-spline curve.
In other words, uniform periodic B-spline is used to model closed curves. Some
of the uniform knot vectors, for example, are shown below.
A knot vector with uniform spacing looks like
[
−
1
.
5
,
−
1
.
0
,
−
0
.
5
,
0
.
0
,
0
.
5
,
1
.
0
,
1
.
5
,
2
.
0]
.
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