Bspline implementation not working as expected - c

Question

I made this implementation of the BSPLINE curve. I Followed the usual definition presented in http://en.wikipedia.org/wiki/B-spline
t is the knot vector.
#include <stdio.h>
double N(int i, int k, double u, double t[])
{
if(k == 1)
{
if(u >= t[i] && u < t[i+1])
return 1.0e0;
else {
return 0.0e0;
}
}
return ((u - t[i])*N(i, k -1, u, t))/(t[i+k-1] - t[i]) + ((t[i+k] - u)*N(i+1, k-1, u, t))/(t[i+k] - t[i+1]);
}
double pu(double u, double x[], int n, int k, double t[])
{
int i;
double r = 0.0e0;
for(i = 0; i < n; i++)
{
r += x[i]*N(i, k, u, t);
}
return r;
}
int main()
{
double t[] = {0.0, 0.5, 1, 2, 3, 4, 4.5, 5}; //knot vector
double x[] = {-30.0, 25.0, 9.0, 20.0, 25.0, 31.0}, y[] = {-5.0, -10.0, 3.0, -10.0, -5.0, 25.0}; //the points
double u;
for(u = 0.0e0; u < 5.0; u+=0.01e0)
{
printf("%lf %lf\n", pu(u, x, 6, 2, t), pu(u, y, 6, 2, t));
}
return 0;
}
The problem is that when I plot the computed points I see that there is an unexpected behavior at the beginning and end of the curve.
For instance:
I can not understand why this happens, try changing the values of t, but it seems that is not it.

Answer 1

There are two implementations of bspline: uniform and standard. In uniform the first and the last control points are not interpolated and in the standard knot sequence both are interpolated. In uniform, you can have uniform knots usually 1,2,3,... For standard knot sequence if you have order k (degree k-1) you should have k zeros, k ones and fill in the middle with 1/(m-k+2) where m is the number of control points. For example, having 5 control points and order 3, knot sequences are 0, 0, 0, 0.25, 0.5, 0.75, 1, 1, 1.
In addition, using delta functions you can have a much better implementation rather than computing N function. Delta function benefits from the local support of bspline. I suggest you take a look at the course notes that I teach at the University of Calgary:
http://pages.cpsc.ucalgary.ca/~amahdavi/pmwiki-2.2.8/uploads/Site/notes1.pdf
Check page 40 algorithm 3.3.
Hopefully it's helpful.

Answer 2

In general: if you have n control points and you construct a bspline curve of degree k, your knot vector has n+k+1 knots. The domain of the curve (i.e. where the summation of the basis functions are =1) is given in the interval [t_k, ... t_n] (as far as my details are right).
In the loop, your parameter values range from 0.0 to 5.0. This should be t[k] = t[2] = 1.0 to t[n] = t[6] = 4.0.
The wiki examples do it the other way, as the knot vectors given there have multiple values (i.e. k-times) at the beginning and the end. So the shown spline curve starts/ends at the first/last control point. Your construction does not give this property.