MATLAB Code Implementation for B-Spline Curve Drawing

B-spline curves are essential curve representation methods in Computer-Aided Geometric Design (CAGD), offering superior local control and flexibility compared to Bézier curves. Implementing B-spline drawing in MATLAB requires mastery of several key technical aspects: knot vector selection, De Boor algorithm implementation, and piecewise curve rendering.

### Knot Vector Selection The properties of B-spline curves largely depend on knot vector distribution. Four common types include: Uniform knot vector: Equally spaced knots, suitable for most conventional applications. Quasi-uniform knot vector: Increased multiplicity at endpoints, ensuring the curve passes through the first and last control points. Non-uniform knot vector: Unequal knot spacing, enabling more flexible curve adjustments. Open uniform knot vector: Combines uniform and quasi-uniform characteristics with identical endpoint multiplicity. When selecting knot vectors, note that the number of knots must satisfy the relationship m = n + p + 1, where n is the number of control points minus 1, and p is the curve degree.

### De Boor Algorithm Implementation The De Boor algorithm is the core method for B-spline curve evaluation, analogous to the de Casteljau algorithm for Bézier curves but incorporating knot interval determination. Key steps include: Identifying the node interval containing parameter u. Calculating intermediate control points through recursive formulas, with each recursion level reducing one control point. Obtaining the curve coordinate at parameter u at the base level. In MATLAB implementation, use loop structures for the recursive process, paying special attention to handling cases with repeated knots. A typical implementation involves creating a temporary control point array that gets updated iteratively through the algorithm's pyramid structure.

### Curve Rendering Complete B-spline curve drawing requires: Sampling parameter u at equal intervals within the domain. Applying the De Boor algorithm for each u value to compute corresponding curve points. Connecting all points to form a smooth curve. For efficiency optimization, precompute knot interval indices to avoid repeated searches during evaluation. When drawing curves of different degrees, adjust the p value to observe changes in curve smoothness. Additionally, when dragging control points, only recalculate affected curve segments locally, demonstrating B-spline's local support property. This can be implemented using MATLAB's callback functions for interactive control point manipulation.

Through this implementation, one can deeply understand B-spline theoretical foundations, including basis function recursive definitions and curve continuity conditions. These concepts are thoroughly derived in CAGD textbooks. During actual programming, pay attention to numerical stability issues, particularly when handling denominator zero situations with repeated knots. Implementing safeguards like epsilon comparisons for floating-point equality checks is crucial for robust code.