Explain about cox deBoor algorithm with equations?

Answer 1

The Cox-de Boor algorithm is a recursive method used for evaluating B-spline basis functions. It is defined as follows:

1. For a non-negative integer ( n ) and a knot vector ( \mathbf{U} ):
   • If ( n = 0 ): [ N_{0,j}(u) = \begin{cases} 1 & \text{if } u_j \leq u < u_{j+1} \ 0 & \text{otherwise} \end{cases} ]
   • If ( n > 0 ): [ N_{n,j}(u) = \frac{u - u_j}{u_{j+n} - u_j} N_{n-1,j}(u) + \frac{u_{j+n+1} - u}{u_{j+n+1} - u_{j+1}} N_{n-1,j+1}(u) ]

This algorithm allows for efficient computation of B-spline basis functions, which are essential in computer graphics and data fitting.