Graphics Reference
In-Depth Information
using the same parameter u at each stage in Step 2 of Algorithm 11.5.2.1, use a dif-
ferent name for the parameter at each stage, so that
p
r
would become a function of
r variables. At the end,
p
n
would be the blossom. For example, in the cubic case we
would have
pp
p
p
0
1
2
3
1
()
1
()
1
()
p
u
p
u
p
u
1
1
1
0
1
2
2
(
)
2
(
)
p
uu
,
p
p
uu
uu u
,
,
12
12
0
1
3
(
)
,
123
0
with
p
3
(u
1
,u
2
,u
3
) the blossom of p(u). Check that this works for the quadratic and
cubic Bézier curve case.
11.5.2.2
Consider the B-spline curve p(u) of order 3 with control points (1,2), (2,3), (5,1), (6,2)
and knot vector (0,0,0,1,4,4,4) defined by equation (11.95). Find the three control
points
q
0
,
q
1
, and
q
2
so that the quadratic Bézier curve q(u) defined on [0,1] with
those control points generates the same curve as p(u)|[1,4].
11.5.2.3
Again consider the quadratic B-spline curve p(u) from Exercise 11.5.2.2. Find the
control points for the B-spline curve that is obtained after inserting the knot t = 2.
Section 11.5.4
11.5.4.1
Prove equation (11.107).
Section 11.9
Consider the curves p: [0,1] Æ
R
2
and q: [0,1] Æ
R
2
defined by
11.9.1
(
)
(
)
()
=-+ -+
2
3
()
=
2
3
p t
u
21
u
,
u
33
u
and
q t
u
,
u
+
1
.
Clearly, p(1) = (0,1) = q(0). Show that the composite curve is C
1
but not G
1
.
11.17
P
ROGRAMMING
P
ROJECTS
1. Display and manipulation of curves (Sections 11.4, 11.5.1, and 11.5.3)
Rather than listing specific projects, we shall simply suggest that the reader implement any one
of the many types of curves we have described in this chapter and provide a user interface that
allows one to manipulate the parameters associated to them. Of special interest are the many
B-spline curves. Allow comparison of the curves by displaying several modifications of a curve
simultaneously. Try to limit the amount of typing one has to do for the program and allow as
much geometric input using the mouse as possible.
2. Interpolation (Sections 11.2.3 and 11.5.5)
Implement some interpolating splines for user picked or defined points.
