Why B-Splines

Bezier curves of degree n are global: moving one control point changes the entire curve. B-Splines decouple degree from number of control points and provide local control - moving one control point only affects a few segments. This is why CAD systems and animation packages standardize on B-Splines (and their rational extension, NURBS).

Definition

P(u) = sum over i=0..n of N_(i,p)(u) * P_i, where N_(i,p)(u) is the B-Spline basis of degree p = k - 1, defined recursively (Cox-de Boor):

• N_(i,0)(u) = 1 if t_i <= u < t_(i+1), else 0.
• N_(i,p)(u) = (u - t_i)/(t_(i+p) - t_i) N_(i,p-1)(u) + (t_(i+p+1) - u)/(t_(i+p+1) - t_(i+1)) N_(i+1,p-1)(u).

Knot Vector

A non-decreasing sequence t = (t_0, t_1, ..., t_(n+k)) of length n + k + 1.

• Uniform: equally spaced (e.g., 0,1,2,3,...).
• Non-uniform: spacing varies for tension control.
• Clamped (open uniform): first and last knots have multiplicity k -> curve interpolates the first and last control points (like Bezier endpoints).

Properties

• Local support: changing one P_i affects only the part of the curve where N_(i,p)(u) > 0, i.e., the interval [t_i, t_(i+p+1)].
• Convex hull within each span: the curve in [t_j, t_(j+1)] lies inside the convex hull of P_(j-p), ..., P_j.
• Continuity at simple knots: C^(p-1).
• Reduces to Bezier when knot multiplicity equals degree at the ends and there are no interior knots.

Cubic B-Spline (p=3, uniform)

With four control points P0..P3 the segment is P(u) = ((1-u)^3/6) P0 + ((3u^3 - 6u^2 + 4)/6) P1 + ((-3u^3 + 3u^2 + 3u + 1)/6) P2 + (u^3/6) P3, which does not interpolate P0 or P3 (clamped knot vectors restore endpoint interpolation).

Comparison: Bezier vs B-Spline

NURBS - Brief

Non-Uniform Rational B-Splines add weights w_i to each control point and evaluate P(u) = (sum w_i N_i,p P_i) / (sum w_i N_i,p). This lets B-Splines exactly represent conic sections (circles, ellipses, parabolas) which polynomial splines cannot. NURBS is the standard for industrial CAD and STEP file exchange.

Practical Tips

• To pull a B-Spline closer to a particular point, increase that point's weight (NURBS) or insert a knot near that location (knot insertion).
• Knot multiplicity p reduces continuity by p, so designers use repeated knots to introduce sharp corners deliberately.