## 1. **NURBS Curve Definition**

A **NURBS curve** of degree \\( p \\) is defined as:


\\[
\mathbf{C}(u) = \frac{\sum_{i=0}^{n} N_{i,p}(u) w_i \mathbf{P}_i}{\sum_{i=0}^{n} N_{i,p}(u) w_i}, \quad u \in [u_0, u_m]
\\]


Where:
- \\( \mathbf{P}_i \\): Control points
- \\( w_i \\): Weights (positive real numbers)
- \\( N_{i,p}(u) \\): B-spline basis functions of degree \\( p \\)
- \\( u \\): Parameter value
- \\( n \\): Number of control points - 1
- \\( m \\): Number of knots - 1
- \\( [u_0, u_m] \\): Domain of the curve (defined by the knot vector)



## 2. **B-spline Basis Functions**

B-spline basis functions are defined **recursively** using the **Cox–de Boor recursion formula**:

### Base case (degree 0):

\\[
N_{i,0}(u) =
\begin{cases}
1 & \text{if } u_i \le u < u_{i+1} \\
0 & \text{otherwise}
\end{cases}
\\]

### Recursive case (degree \\( p > 0 \\)):
\\[
N_{i,p}(u) = \frac{u - u_i}{u_{i+p} - u_i} N_{i,p-1}(u) + \frac{u_{i+p+1} - u}{u_{i+p+1} - u_{i+1}} N_{i+1,p-1}(u)
\\]

(With the convention that any term with a zero denominator is taken to be zero.)