Graphics Reference
In-Depth Information
6.3.1 Shape Parameters
Barsky also examined the conditions for continuous shape parameters. Let
β
1
,i
(
t
)and
β
2
,i
(
t
) be the values of the shape parameters at the point
P
i
(
t
),
where
i
=1
,
2
,
,m
. Hence, we can choose different values of shape pa-
rameters to exhibit the local behavior of the curve. Now, if we consider a
complete curve consisting of many pieces, then at each joint between two
such pieces, shape parameters should have unique values, i.e., at
P
i
+1
(0) =
P
i
,i
=1
,
2
,
···
···
,m
−
1wehave,
β
1
,i
+1
(0) =
β
1
,i
(1)
,
β
2
,i
+1
(0) =
β
2
,i
(1)
,
i
=1
,
2
,
···
,m
−
1
.
When an user specifies the values of
β
1
and
β
2
, it can be taken as discrete
parameter values. We represent the discrete values of
β
1
,i
and
β
2
,i
as
α
1
,i
and
α
2
,i
respectively. Therefore, we write,
β
1
,
1
(0) =
α
1
,
0
and
β
2
,
1
(0) =
α
2
,
0
.Thus,
between two pieces of curves we can write,
β
1
,i
+1
(0) =
α
1
,i
=
β
1
,i
(1)
,
β
2
,i
+1
(0) =
α
2
,i
=
β
2
,i
(1)
,i
=1
,
2
,
···
,m
−
1
.
Hence,
α
1
,m
=
β
1
,m
(1) and
α
2
,m
=
β
2
,m
(1). A solution for this is as follows:
β
1
,i
(
t
)=(1
−
t
)
α
1
,i−
1
+
tα
1
,i
,
(6.38)
β
2
,i
(
t
)=(1
−
t
)
α
2
,i−
1
+
tα
2
,i
,i
=1
,
2
,
···
,m
−
1
.
In addition,
delta
=2
β
1
+4
β
1
+4
β
1
+
β
2
+ 2 becomes
δ
i
(
t
)=2
β
1
,i
(
t
)+4
β
1
,i
(
t
)+4
β
1
,i
(
t
)+
β
2
,i
(
t
)+2
.
Finally, the discrete analogue to
δ
i
(
t
)is
γ
0
=
δ
1
(0)
,
γ
i
=
δ
i
(1)
.
6.3.2 End Conditions of Beta Spline Curves
Suppose we have
m
+1 control vertices, say
V
0
,V
1
,V
2
,
···
,V
m
. Then the control
polygon defined by these vertices help to generate
m
−
2 pieces of a complete
curve curve, namely,
P
2
(
t
),
P
3
(
t
),
···
,
P
m−
1
(
t
). The
β
-spline curve starts at
P
2
(0) = (2
α
1
,
0
V
0
+(
γ
1
−
2
α
1
,
0
−
2)
V
1
+2
V
2
)
/γ
1
(6.39)
and ends at
P
m−
1
(1) = (2
α
1
,m
V
m−
2
+(
γ
m−
1
−
2
α
1
,m
−
2)
V
m−
1
+2
V
m
)
/γ
m−
1
.
(6.40)
In a real situation we should have the objective for the curve to start at
V
0
and end at
V
m
. This is accomplished through the use of multiple vertices as
well as by phantom vertices.
Search WWH ::
Custom Search