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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
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 ) 0 ,
P m (1) = 6 α 1 ,m ( α 1 ,m + 1)( V m
V m− 1 ) 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 ) 0 ,
P m (1) = 6(2 α 1 ,m
2 α 1 ,m
α 2 ,m )( V m
V m− 1 ) 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 ) 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 2) u m
S i,j ( u, v )=
×
n = 2
m =0
(6.43)
q =1
p =3
e pq ( β 1 2) v p V i + n,j + q .
q = 2
p =0
Rearranging, we get
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