Drawing NURBS with MATLAB: Implementation and Code Techniques

When plotting NURBS (Non-Uniform Rational B-Spline) curves in MATLAB, the process requires defining core parameters including control points, weights, knot vectors, and basis function calculations. While MATLAB doesn't provide dedicated NURBS functions out-of-the-box, implementation can be achieved through fundamental curve generation principles or third-party toolboxes like the NURBS Toolbox.

Implementation Approach Defining Control Points and Weights: The shape of NURBS curves is determined by control points, where each point is associated with a weight value that influences how closely the curve approximates the control point. In code, this typically involves storing control points as a matrix and weights as a vector. Constructing Knot Vectors: Knot vectors define the parametric space segmentation and must satisfy non-decreasing order with appropriate multiplicity rules (e.g., for open curves, end knots typically have multiplicity equal to the curve degree). Implementation requires careful validation of knot sequence properties. Calculating Basis Functions: Using recursive algorithms like the Cox-de Boor algorithm to compute B-spline basis functions, then combining them with weights to generate rational basis functions. This recursive computation can be implemented through nested loops or vectorized operations. Parametric Sampling: Uniformly sample parameter values within the domain, compute curve coordinates point-by-point using the basis functions and control points, and connect them to form the curve. The sampling density affects the smoothness of the final visualization.

Extended Considerations For higher efficiency or complex operations like surface modeling, consider using dedicated NURBS libraries or converting to MATLAB's built-in B-spline functions such as `spcol` or `bspline` related functions. These functions can handle basis function computations more efficiently. For non-closed curves, verify the multiplicity of endpoints in the knot vector to prevent abnormal behavior at curve boundaries. Code should include checks for knot vector validity before curve computation.

Following these steps enables basic NURBS visualization without relying on external toolboxes, providing flexibility for custom implementations and algorithm modifications.