Computational geometry 135
The derivatives with respect to the Lagrange multipliers λ
l
produce
(
n
i=0
B
i,k
(t
l
)Q
i
− R
l
)=0;l =0, 1,...,o.
This results in o + 1 new equations of the form
n
i=0
B
i,k
(t
l
)Q
i
= R
l
,
or in matrix form, using the earlier matrices:
MBQ = R,
where R is a vector of the interpolated points. The two sets of equations may
be assembled into a single matrix equation with n+1+o+1 rows and columns
of the form
AB
T
M
T
MB 0
Q
Λ
=
B
T
P
MP
.
The first matrix row represents the constrained functional and the second
row represents the constraints. The simultaneous solution of this (symmetric,
indefinite, but still linear) system produces the optimized (approximated and
smoothed) and selectively interpolated solution.
The solution of this problem is accomplished ...