B-Spline Construction

De Boor’s algorithm, a maestro of basis spline refinement, meticulously navigates through knots, unraveling the intricacies of B-splines with mathematical precision. Meanwhile, De Casteljau, the geometric orchestrator, takes center stage in the Bezier ballet, elegantly guiding control points through a recursive dance. Together, these algorithms fuse art and mathematics, seamlessly sculpting curves and splines with technical finesse, creating a harmonious symphony of numerical intricacies in computational geometry. ChatGPT is so emotional today. I have been studying the Bezier-De Casteljau algorithm for several years. Last week, I finally managed to understand and implement B-Spline Construction in Rhino and Grasshopper Python code, using the Cox-De Boor algorithm. This algorithm calculates the effect of control points on a B-Spline with many control points. After this, I am not limited to the degree+1 control points. This animation shows both algorithms working together.

This Rhino Python code in a Grasshopper definition can handle three tasks. The first code draws a B-Spline with given control points and knot vector and degree. The knot vector is a list of numbers that controls many aspects of the process. The second code imitates the parametric evaluation so that a single point can be located on the curve. Finally, the third code handles and Bezier-de Casteljau algorithm and locates points on the Bezier curves with cp-1 degree.