B-splines: local control, and why CAD uses them

Take the Bézier idea and raise the point count. With 10 control points you get a degree-9 polynomial, and something unpleasant happens: every point influences every part of the curve. Drag one point at the far left and the far right end moves too. That is called *non-local* control, and it makes high-degree Béziers useless for design.

A B-spline keeps the same "weighted blend of nearby points" idea but makes the blend depend on a knot vector — a non-decreasing list of parameter values. Each control point has a limited interval where its weight is non-zero. That gives local control.

// Degree p=3 (cubic). Knot vector for a clamped curve with n+1 points:
//   [0,0,0,0, u1, u2, ..., un-3, 1,1,1,1]
//   ^-- repeated p+1 times at each end  = clamped: curve touches both ends

// Cox-de Boor recursion — the standard way to evaluate it
//   Base:      N_{i,0}(u) = 1 if u_i <= u < u_{i+1} else 0
//   Recurse:   N_{i,p}(u) = ((u-u_i)/(u_{i+p}-u_i))       * N_{i,p-1}(u)
//                         + ((u_{i+p+1}-u)/(u_{i+p+1}-u_{i+1})) * N_{i+1,p-1}(u)
//
// The curve is then:  C(u) = sum_i  N_{i,p}(u) * P_i

Two properties explain why B-splines are the industry default:

Local support. Moving one control point changes only a bounded stretch of the curve. You can edit a car door without disturbing the roof.

Convex hull property. Each segment lies inside the hull of its neighbouring points. That makes collision and intersection tests tractable — you get cheap bounds for free.