where Bspline curve is not continuous

Question

Consider the (infinite) knot vector $ \tau $ := ($t_0, t_1, t_2, t_3, t_4, ...$) with $t_0=0, t_1=1, t_2=t_3=2\ and\ t_j = j-1 $ for all $\ j \in N\backslash \{ 1,2,3 \} $. Identify all (permissible) values for $i \in N\ where\ N=\{0,1,2,3,...\} $ such that $N_i,_1,_\tau(t) $ is not continuous.

would you please give me hint regarding above simple question! (thank you in advance)

here I got stuck :

$N_{i,1}(t)=\frac{t -t_{i}}{t_{i+1}-t_{i}}N_{i,0}(t)+ \frac{t_{i+2}-t}{t_{i+2}-t_{i+1}}N_{i+1,0}(t)$ .......... (1)

$N_{0,1}(t)=\frac{t -t_{0}}{t_{1}-t_{0}}N_{0,0}(t)+ \frac{t_{2}-t}{t_{2}-t_{1}}N_{1,0}(t)$ .......... (2)

$ because : t_{1}- t_{0} = 1 , t_{2}- t_{1} = 1 , t_{3}- t_{2} = 0 , t_{4}- t_{3} = 1 ,..., thus\ they\ are\ not\ uniform\ !? ... $

$N_{0,1}(t)=\frac{t}{1}N_{0,0}(t)+ \frac{2-t}{1}N_{1,0}(t)$ .......... (3)

$N_{1,1}(t)=\frac{t-1}{1}N_{0,0}(t)+ \frac{2-t}{0}N_{1,0}(t)$ .......... (4)

so second part is undefined because ($\frac{2-t}{0}$), Can I say $N_i,_1,_\tau(t) $ is not continuous only in $ [t_{3} , t_{4}[ ? $

Answer 1

This is a very easy question. Just use the definitions and what you know.

1. In the given knot sequence, what are the multiplicities of the various knots?
2. What are the degrees of the basis functions you're being asked about?

From (1) and (2), what can you conclude about the continuity of the basis functions?

All the algebra you did with the recursive definitions of the basis functions is unnecessary. Just use the fact that you quoted in your comment above.

Answer 2

the sequence of (infinite) knot vector is $\tau = (0,1,2,2,3,4,5,6,...)$

multiplicity for $t_{0} ⇒ multiplicity\ 1 , $
$ k\ −\ r\ times\ continuously\ differentiable\ at\ a\ knot\ of\ multiplicity\ r $

$ ⇒\ k=1\ ,\ r=1\ at\ t_{0}\ is\ k\ -\ r\ = 1 - 1 = 0 \\$

at $t_{1} ⇒ multiplicity\ is\ 1 \ ,\ k - r = 1-1 = 0 $ $ \_ $

at $t_{2} , t_{3} ⇒ multiplicity\ is\ 2\ ,\ k - r = 1-2 = < 0 \ !!!!!! \\ $ $ \_ $

I know if multiplicity is equal to degree then we can say Bspline basis function is continuous, but what about the above case where K < r ??????

at $t_{4} ⇒ multiplicity\ is\ 1,\ k - r = 1-1 = 0 \\ $ $ \_ $ and so on ....