Solving a B-spline exercise

2012-07-26T09:02:41Z

Hello,

I'm trying to solve the B-spline equation for a given set of variables but I can never arrive at the answer in the book. I think there is some core concept I'm not understanding.

Here are the variables $n = 3 \mbox{, }p = 1 \mbox{, } u=1.5 \mbox{ and } U = \{0,1,2,3\}$

This is the equation in the theory:

$B _{i , 0} (u ) = \left\{ \begin{array}{rcl} 1 & \mbox{ if } u_i ≤ u < u_{i +1} \\ 0 & \mbox{otherwise} \end{array}\right.$

$B _{i , p} (u ) = \frac{u − u_i}{u_{i + p} − u_i}B _{i , p-1} (u )-\frac{u_{i+p+1} − u}{u_{i + p+1} − u_{i+1}}B _{i+1 , p-1} (u )$

The values I'm asked to compute and their answer:
$B _{0 , 1} (1.5 ) =0$
$B _{1 , 1} (1.5 ) =0.5$
$B _{2 , 1} (1.5 ) =0.5$
$B _{3 , 1} (1.5 ) =0$

For the first value I'm doing:
$B _{0 , 0} (1.5 ) =0 \mbox{ and } B _{1 , 0} (1.5 ) =1$
$B _{0 , 1} (1.5 ) = \frac{1.5 − 0}{1 − 0}B _{0 , 0} (1.5 )-\frac{2 − 1.5}{2 − 1}B _{1 , 0} (1.5 )=1.5*0+0.5*1=0.5$
That doesn't match the above answer.

In the second value I also end up with 0.5.
In the third I don't understand how to get $u _{4}$ when the U vector only has 4 knots.
In the forth is the same problem.

Is this exercise wrong or did I get it wrong?

Solving a B-spline exercise - MathOverflow

Solving a B-spline exercise