Graphics Reference
In-Depth Information
End Vertex Interpolation
It is convenient as well as meritorious to start the curve at
V
0
and end at
V
m
. It is a special case of the previous end conditions where
P
0
=
V
0
and
P
m
=
V
m
. From equation (6.41), phantom vertices can be written as
(
α
1
,
0
+1)
V
0
−
/α
1
,
0
,
V
−
1
=
{
V
1
}
V
m
+1
=(
α
1
,m
+1)
V
m
−
α
1
,m
V
m−
1
.
Values of the first derivative vectors at each end point are
P
1
(0) = 6(
α
1
,
0
+ 1)(
V
1
−
V
0
)
/γ
0
,
P
m
(1) = 6
α
1
,m
(
α
1
,m
+ 1)(
V
m
−
V
m−
1
)
/γ
0
.
This shows that the curve is tangent to the control polygon at each end point.
Substitution into equation (6.42) gives
P
1
(0) = 6(2
α
1
,
0
+
α
2
,
0
−
2)(
V
1
−
V
0
)
/γ
0
,
P
m
(1) = 6(2
α
1
,m
−
2
α
1
,m
−
α
2
,m
)(
V
m
−
V
m−
1
)
/γ
m
.
Hence, at the initial point of the curve, the first and second derivative vectors
are related as
P
1
(0) =
P
1
(0)
,
(2
α
1
,
0
+
α
2
,
0
−
{
2)
/
(
α
1
,
0
+1)
}
P
m
(1) =
P
m
(1)
.
(2
α
1
,m
−
2
α
2
,m
)
/α
1
,m
(
α
1
,m
+1)
{
2
α
1
,m
−
}
Assuming distinct vertices, the first and second derivatives are non-zero, and
the first and second derivative vectors are linearly dependent at the initial
and final points of the curve.
6.4 Beta-Spline Surface
A
β
spline surface is a straightforward extension of the
β
spline curve in two
dimensions. Mathematically, it is the Cartesian cross product of two sets of
orthogonal curves. The (
i, j
)
th β
-spline surface patch is given by
n
=1
m
=3
c
mn
(
β
1
,β
2)
u
m
S
i,j
(
u, v
)=
×
n
=
−
2
m
=0
(6.43)
q
=1
p
=3
e
pq
(
β
1
,β
2)
v
p
V
i
+
n,j
+
q
.
q
=
−
2
p
=0
Rearranging, we get
Search WWH ::
Custom Search