Need help with B-spline determination
Hi, i got problem in solving one task namely:
I need to determine the knot vector and the base for B-spline of degree 3 , N0,3(u) for all the intervals determined with the knots (the intervals are all with positive numbers), if the base is uniform and the vector of knots is uniform as well with starting value (knot) u0 = 0 ,and all are apart of 1 so next knot is on distance 1 from the previous and so on..
I started this way , tell me if im wrong somewhere:
Because the control points are not given nor their number, i derived this formula for the size of the vector: m = n+3+1 = n+4, so i must have at most n+4 knots in the vector and the vector would look like [u0, u1, ...., um] ui-u(i+1) = 1 , that is they are distanced 1 for every adjacent knot, so i assume my formula is correct there for the number of knots, next i dont quite understand what is asked from me to find for N0,3(u) but i started it this way:
Ni,1(u)=1 if u is between knots i and i+i, and 0 otherwise, and for the recursive part i have : Ni,p(u) = (u-xi)*Ni,p-1(u)/(xi+p-1 - xi) + (xi+p-u)*Ni+1,p-1(u)/(xi+p - xi+1), which is base for the i-th knot , the actual de-boor or what his name was algorithm, so for N0,3(u) = N0,3(u) = (u-x0)*N0,2(u)/(x2 - x0) + (x3-u)*N1,2(u)/(x3 - x1), where u should be between xi and xi+i or in this case is between x0 and x1, so i can draw the recursion tree for the calls of the bases and ill draw the recursions till i reach base of order p=1, which trivially is solved, either is 1 or 0..
I dont know how near is my solution to this problem but im kinda messed up with the b-splines as i cant find really step by step guide im a bit confused with the knot and control points relation , the cubic spline is much easier for me and also the bezier is quite easy but this b-splines stuff i dont quite understand, and i also dont know what's the difference between uniform base and non uniform base, i know about the vectors of knots for uniform i got equidistance between the knots and for non uniform there might be same or not distances between points..
Thanks for help! (Itwasntme)