Construction of the basis functions Non-uniform rational B-spline

from top bottom: linear basis functions




n

1
,
1




{\displaystyle n_{1,1}}

(blue) ,




n

2
,
1




{\displaystyle n_{2,1}}

(green) (top), weight functions



f


{\displaystyle f}

,



g


{\displaystyle g}

(middle) , resulting quadratic basis function (bottom). knots 0, 1, 2, , 2.5








n

i
,
n


=

f

i
,
n



n

i
,
n
−
1


+

g

i
+
1
,
n



n

i
+
1
,
n
−
1




{\displaystyle n_{i,n}=f_{i,n}n_{i,n-1}+g_{i+1,n}n_{i+1,n-1}}

f

i




{\displaystyle f_{i}}

rises linearly 0 1 on interval




n

i
,
n
−
1




{\displaystyle n_{i,n-1}}

non-zero, while




g

i
+
1




{\displaystyle g_{i+1}}

falls 1 0 on interval




n

i
+
1
,
n
−
1




{\displaystyle n_{i+1,n-1}}

non-zero. mentioned before,




n

i
,
1




{\displaystyle n_{i,1}}

triangular function, nonzero on 2 knot spans rising 0 1 on first, , falling 0 on second knot span. higher order basis functions non-zero on corresponding more knot spans , have correspondingly higher degree. if



u


{\displaystyle u}

parameter, ,




k

i




{\displaystyle k_{i}}





i


{\displaystyle i}

knot, can write functions



f


{\displaystyle f}

,



g


{\displaystyle g}

as

f

i
,
n


(
u
)
=



u
−

k

i





k

i
+
n


−

k

i







{\displaystyle f_{i,n}(u)={{u-k_{i}} \over {k_{i+n}-k_{i}}}}

and

g

i
,
n


(
u
)
=




k

i
+
n


−
u



k

i
+
n


−

k

i







{\displaystyle g_{i,n}(u)={{k_{i+n}-u} \over {k_{i+n}-k_{i}}}}

the functions



f


{\displaystyle f}

,



g


{\displaystyle g}

positive when corresponding lower order basis functions non-zero. induction on n follows basis functions non-negative values of



n


{\displaystyle n}

,



u


{\displaystyle u}

. makes computation of basis functions numerically stable.

again induction, can proved sum of basis functions particular value of parameter unity. known partition of unity property of basis functions.

the figures show linear , quadratic basis functions knots {..., 0, 1, 2, 3, 4, 4.1, 5.1, 6.1, 7.1, ...}

one knot span considerably shorter others. on knot span, peak in quadratic basis function more distinct, reaching one. conversely, adjoining basis functions fall 0 more quickly. in geometrical interpretation, means curve approaches corresponding control point closely. in case of double knot, length of knot span becomes 0 , peak reaches 1 exactly. basis function no longer differentiable @ point. curve have sharp corner if neighbour control points not collinear.