Derive the Non Uniform Rationlize B- Spline( NURB) curve

To derive a Non-Uniform Rational B-Spline (NURBS) curve, we need to understand the basic concepts behind B-splines, rational curves, and how they come together in NURBS. I'll walk through the key steps involved in deriving a NURBS curve.

### 1. **B-Spline Curves**
A B-spline curve is defined by a set of control points, a degree \( p \), and a knot vector. The B-spline curve of degree \( p \) can be expressed as:

\[
\mathbf{C}(u) = \sum_{i=0}^{n} N_{i,p}(u) \mathbf{P}_i
\]

Where:
- \( \mathbf{C}(u) \) is the B-spline curve.
- \( \mathbf{P}_i \) are the control points.
- \( N_{i,p}(u) \) are the B-spline basis functions of degree \( p \).
- \( u \) is the parameter along the curve.
- \( n \) is the number of control points minus 1.

### 2. **B-Spline Basis Functions**
The B-spline basis functions \( N_{i,p}(u) \) are recursively defined using the knot vector \( \{u_0, u_1, \ldots, u_{m}\} \) as follows:

- **Degree 0** basis functions:
  \[
  N_{i,0}(u) = 
  \begin{cases} 
  1 & \text{if } u_i \leq u < u_{i+1} \\
  0 & \text{otherwise}
  \end{cases}
  \]

- **Degree \( p \) basis functions** for \( p > 0 \):
  \[
  N_{i,p}(u) = \frac{u - u_i}{u_{i+p} - u_i} N_{i,p-1}(u) + \frac{u_{i+p+1} - u}{u_{i+p+1} - u_{i+1}} N_{i+1,p-1}(u)
  \]

### 3. **Rational B-Spline Curves**
A rational B-spline curve introduces weights to each control point, resulting in a weighted combination of the basis functions. The curve is defined as:

\[
\mathbf{R}(u) = \frac{\sum_{i=0}^{n} N_{i,p}(u) w_i \mathbf{P}_i}{\sum_{i=0}^{n} N_{i,p}(u) w_i}
\]

Where:
- \( w_i \) are the weights associated with each control point \( \mathbf{P}_i \).
- \( \mathbf{R}(u) \) is the rational B-spline curve.

### 4. **NURBS Curve**
Finally, a NURBS curve is simply a rational B-spline curve, where the basis functions \( N_{i,p}(u) \) are the same as those of a non-rational B-spline. The NURBS curve is given by:

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

This equation combines the control points, basis functions, and weights into a single formula, resulting in a powerful tool for representing complex shapes. The knot vector controls the parameterization of the curve, influencing how the curve interpolates or approximates the control points.

### Summary
- **Control Points \( \mathbf{P}_i \)**: Define the shape of the curve.
- **Knot Vector**: Defines the parameter space and influences the curve's smoothness.
- **Weights \( w_i \)**: Allow for rational representation, enabling conic sections and more complex shapes.

### Tip
In practice, choosing an appropriate knot vector and weights is crucial for achieving the desired curve shape in NURBS modeling.

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Would you like more details or have any specific questions? Here are five related questions:

1. How do the weights \( w_i \) affect the shape of a NURBS curve?
2. What is the role of the knot vector in NURBS?
3. How can a NURBS curve represent a circle or other conic sections?
4. Can you explain the difference between uniform and non-uniform B-splines?
5. How do NURBS curves compare to Bezier curves in terms of flexibility?

**Tip:** Always ensure that the sum of the weights in a NURBS curve is non-zero to avoid undefined points.

Learn how to derive a Non-Uniform Rational B-Spline (NURBS) curve step-by-step. Understand the concepts of B-spline curves, rational B-spline curves, and how they combine into NURBS. Suitable for advanced mathematics enthusiasts and students.

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