SPLINES 145
t
t
ii
t
t
i i i
i
i
i
i
MtdtMt dtMtt t
–1 –1
∫∫1, () = 1, () = 1,()( – –1) = 1
or Mti tt
i i
1,
–1
() =^1
- for t∈ [ti–1,ti); (5.25)
= 0 elsewhere
Combining Eqs. (5.24) and (5.25), the recursion relation for a B-spline basis function may be
written as
Mti tt
i i
1,
–1
() = –^1 for t∈ [ti–1,ti);
= 0 elsewhere
Mk,i(t) =
tt
tt
Mt
tt
tt
ik Mt
i ik
ki
i
i ik
ki
- () +
- () +
- ()
–1, –1
–1, for t∈ [ti−k,ti); (5.26)
= 0 elsewhere
5.6.1 Normalized B-Spline Basis Functions
The normalized B-spline weight Nk,i(t), which are used more frequently in the design of B-spline
curves may be computed as
Nk,i(t) = (ti–ti–k)Mk,i(t) (5.27)
From Eqs. (5.25) and (5.27), N1,i(t) = (ti–ti–1)M1,i(t) = 1 for t in the range [ti–1,ti) and N1,i(t) = 0 for
all other values of t. We may combine the two results as N1,i(t) = δi, where δi = 1 for t∈ [ti–1,ti) and
δi = 0 elsewhere. For higher order normalized B-splines, the recursion relation can be derived using
Eqs. (5.24) and (5.27). Starting with (5.24)
Mt
tt
tt
Mt
tt
tt
ki ik Mt
i ik
ki
i
i ik
, ki
- –1, –1
- –1, –1
() = –1,
- () +
- () +
()
Table 5.2 Recursion to compute B-spline basis functions
[ti–k,ti–k+1) M1,i–k+1(t)
M2,i–k+2(t)
[ti–k+1,ti–k+2) M1,i–k+2(t)
M2,i-k+3(t)
[ti–k+2,ti–k+3) M1,i–k+3(t)
Mk–1,i–1(t)
M... Mk,i(t)
Mk–1,i(t)
[ti–3,ti–2) M1,i–2(t)
M2,i–1(t)
[ti–2,ti–1) M1,i–1(t)
M2,i(t)
[ti–1,ti) M1,i(t)