# where Bspline curve is not continuous

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 $[t_{3} , t_{4}[ ?$

• What do you know about continuity of b-splines at knots? Do you know how continuity at a knots is related to its multiplicity? – bubba Dec 20 '17 at 12:24
• @bubba : not much only following defination: All B-spline basis functions of degree k are k − r times continuously differentiable at a knot of multiplicity r , and k − 1 times continuously differentiable everywhere else. also first derivation is given in following link (math.stackexchange.com/questions/2570993/…) – GreenQuestioner Dec 20 '17 at 22:13
• @bubba : I added my little solution to above question, thank you for your help, I will be glad to see the proper solution. – GreenQuestioner Dec 20 '17 at 23:17
• @bubba : please look at next question if you are interested (math.stackexchange.com/questions/2570993/…) – GreenQuestioner Dec 20 '17 at 23:22
• What is $N_{i,1,r}$ in the last line of the question ?? – bubba Dec 21 '17 at 0:47

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 ....

• Good progress. If you can't see the multiplicities of $t_4, t_5, \ldots$, then you need to go look at the definition of "multiplicity". – bubba Dec 21 '17 at 10:57
• multiplicity of $t_{4}\ is\ 1$,$t_{5}\ also\ is\ 1$ and so on ... Am I correct ? – GreenQuestioner Dec 21 '17 at 12:28
• Yes, of course. The multiplicity of a knot is just the number of times it occurs in the knot sequence. I'd really like to know what's difficult or confusing about this. Can you tell me, please. – bubba Dec 21 '17 at 12:56
• I know if multiplicity (r) is equal to the degree (k) then we can say Bspline basis function is continuous, but what about the above case where K < r ?????? – GreenQuestioner Dec 21 '17 at 13:30
• @bubba , at $t_{2} , t_{3} ⇒ multiplicity\ is\ 2\ ,\ k - r = 1-2 = < 0 \ !!!!!! \\$ now k< r (degree < multiplicity thus k - r < 0 ) , perhaps this calculation leads me to non-continuity case. Am I correct ? is there any reference or rational explanation? – GreenQuestioner Dec 21 '17 at 14:14

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.