Graphics Reference
In-Depth Information
Each segment of the polygon, except the first and last, is just the graph of the sum
of two of the hat functions. More precisely,
()
=
()
+
()
S t
y b t
y
b
t
for
0
<<
i
n
.
(11.65)
i
i
i
i
+11
i
See Figure 11.12(b). In other words, if we stay away from the endpoints, then
n
Â
()
=
()
S t
y b t
for
x
££
-
t
x
.
(11.66)
ii
1
n
1
i
=
0
Furthermore, the natural parameterization of the graph of a function and equation
(11.66) leads to the following parameterization of
X
:
n
n
Ê
Á
ˆ
˜
=
Â
Â
(
()
)
=
()
()(
)
t
Æ
t St
,
t
,
ybt
bt xy
,
.
ii
i
i
i
i
=
0
i
=
0
The last equality is justified by the fact that
n
Â
0
()
t
=
bt x
,
i
i
i
=
which follows from equation (11.66) applied to the special case where y
i
= x
i
, that is,
the case S(t) = t. In other words, we have another example of a parametric curve in
the form
n
Â
()
=
()
pu
b u
p
,
(11.67)
i
i
i
=
0
for some functions b
i
(u) and “control points”
p
i
. Basically much of the discussion in
this chapter revolves around these are the types of parametric curves and deciding
what is the best choice for the functions b
i
(u).
Using linear B-splines gave us a continuous curve. If we want a smoother curve,
then we need to use higher-degree basis functions. Quadratic B-splines will give us
differentiable curves. Cubic B-splines will give us twice differentiable curves. Here are
the desirable properties that a general B-spline of degree m should have:
(1) It should be a spline of degree m.
(2) It should vanish outside of m + 1 sequentially contiguous spans. Equivalently,
its support should be contained in m + 1 contiguous spans.
We are not going to make this into a formal definition of a B-spline, however, because
we shall reserve that word for a specific family of functions, namely, the function
N
i,k
(u), which will be defined shortly, and those functions will also not be quite as dif-
