Graphics Reference
In-Depth Information
11.3.2
Consider the geometric coefficient matrix
13
1
Ê
ˆ
Á
Á
˜
˜
73
1
B
h
=
18
0
40
Ë
18
0
-
40
¯
for a curve p(u). Analyze p(u) geometrically like we did in Example 11.3.1. In particu-
lar, try to sketch the curve without computing its analytic formula.
11.3.3
Check that formula (11.42) for the interpolating cubic curve works when
p
0
= (0,0),
p
1
= (2,0),
p
2
= (3,2), and
p
3
= (2,3).
11.3.4
Find the matrix
M
for cubic curves p(u) with domain [0,2] that corresponds to the
Hermite matrix
M
h
for curves with domain [0,1] so that the equation
()
()
¢
()
¢
()
p
p
p
p
0
2
0
2
Ê
ˆ
Á
Á
˜
˜
pu
()
=
UM
Ë
¯
holds for all such curves.
Section 11.4
11.4.1
Show that the Bézier basis functions B
i,n
(u) satisfy the recurrence relation
Bu
Bu
()
=
()
=-
1
1
00
,
(
)
(
)
+
()
uB
uuB
u
,
0
££
i
nn
,
>
0
.
in
,
in
,
-
1
i
-
1
,
n
-
1
11.4.2
Let p(u) be a Bézier curve.
(a)
Sketch the value p(2/3) if the control points are (0,0), (2,3), (6,2).
(b)
Sketch the value p(3/4) if the control points are (0,0), (2,3), (6,2), (7,5).
(c)
Sketch the value p(3/4) if the control points are (0,0), (2,3), (2,3), (6,2), (7,5).
Section 11.5.1
11.5.1.1
Compute the B-spline of order 3 that is defined by equation (11.72) if it has control
points (0,0), (2,3), (6,2), (7,5) and knot vector (1,2,4,5,7,8,9).
11.5.1.2
Prove equation (11.74).
Section 11.5.2
11.5.2.1
The blossom of a Bézier curve p(u) with control points
p
i
can be obtained using the
following generalization of the de Casteljau algorithm (see [Rams89]): Rather than
