Graphics Reference
In-Depth Information
We shall look at all of these approaches, except for the second, but begin by looking
at a special case to help clarify and motivate the general discussion later on.
Polygons are the simplest examples of B-spline curves, although one normally
does not think of them that way. Because they are so simple they are useful in high-
lighting some basic aspects of B-splines. Consider a polygon
X
with vertices
p
i
= (x
i
,y
i
),
i = 0,1, . . . ,n. Define functions S
i
and S by
()
=-
(
)
St
1
ty
+
ty
(11.61)
i
i
i
+
1
and
()
=
St
y
ift
<
x
0
0
(
(
)
(
)
)
=
Stx x
-
-
x
fx t x
££
1
,
i
i
i
+
1
i
i
i
+
=
y
if x
<
t
.
(11.62)
n
n
The function S is then a spline function with respect to the knots t
i
= x
i
and the polygon
is just the graph of this function over the interval [x
0
,x
n
]. The graph of S
i
over [x
i
,x
i+1
]
is just the segment
X
i
= [
p
i
,
p
i+1
] of
X
and S
i
corresponds to a linear parameterization
of
X
i
.
Note that if we move any of the points
p
i
, then only the two adjacent functions
S
i-1
and S
i
are affected. We can write S as a sum of basis functions that localize
changes similar to what we saw in the case of Lagrange and Hermite interpolation.
Namely, for each i consider the unit
“hat” function
b
i
(t) defined by
()
=
bt
0
ift x
<
i
i
-
1
(
)
(
)
=-
t
x
x
-
x
if x
££
t
x
,
i
-
1
i
i
-
1
i
-
1
i
(
)
(
)
=
x
-
t
x
-
x
if x
££
t
x
,
i
+
1
i
+
1
i
i
i
+
1
=
0
if x
<
t
.
(11.63)
i
+
1
See Figure 11.12(a). The b
i
are what are called
linear
B-splines because they are
splines that are nonzero on only two spans. A property of these hat functions, which
may not seem important at the moment, is that they sum to 1, that is,
n
Â
()
=
b t
1
for x
£
t
£
x
(11.64)
i
1
n
-
1
i
=
0
Figure 11.12.
Linear B-splines.
