
130 Applied calculus of variations for engineers
the derivative for k =2as
d
dt
B
i,2
(t)=B
i,2
(t)=
2
t
i+2
− t
i
B
i,1
−
2
t
i+3
− t
i+1
B
i+1,1
.
By induction, for any k value the first derivative is as follows:
d
dt
B
i,k
(t)=B
i,k
(t)=
k
t
i+k
− t
i
B
i,k−1
(t) −
k
t
i+k+1
− t
i+1
B
i+1,k−1
(t).
A repeated application of the same step will produce the needed second deriva-
tive B
as
d
dt
B
i,k
(t)=B
i,k
(t)=
k
t
i+k
− t
i
B
i,k−1
(t) −
k
t
i+k+1
− t
i+1
B
i+1,k−1
(t).
The spline, besides being smooth (minimal in curvature), is expected to ap-
proximate a given set of points P
j
; j =0,...,m,with associated prescribed pa-
rameter values (not necessarily identical to the knot values) of t
j
; j =0,...,m.
If suc ...