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(b) Under what conditions does the tangent direction
T
for the curve in-
,2
terpolating P and
P
match the tangent direction
T
for the curve in-
i
+
1
i
+
1,1
terpolating
P
and
P
?
i
+
1
i
+
2
(c)
Find the basis matrix
M
corresponding to the curve interpolating P
KB
and
P that describes the Kochanek-Bartels blending functions. As-
sume that the geometry matrix is
i
+
1
]
G PPPP . [ Hint.
Use a method similar to that which produces the Catmull-Rom basis
matrix in Equation (11.46).]
=
[
KB
i
1
i
i
+
1
i
+
2
()
4. Let
u
be a nonuniform B-spline lying in the x - y plane having control
Q
points
P
=
0, 0
,
P
=
1, 2
,
P
=
2, 2
, and 3
P
=
3, 0
. Suppose the knot
0
1
2
vector is
0, 0, 0, 0,1,1,1,1 . Use Böhm subdivision to insert a new knot at
{
}
P through
′ =
and determine the new control points 0
P .
t
1
2
()
()
5. Calculate the curvature
κ t and the torsion
τ t of the helix given by
()
P
t
=
r
cos ,
t r
sin ,
t ct
.
()
having C continuity, show that
6.
Given a path
P
t
2
3
d
d
d
[
]
2
()
()
()
()
()
P
t
P
t
×
P
t κ t τ t
=
,
2
3
ds
ds
ds
()
()
where
κ t is the curvature of the path and
τ t is the torsion of the path.