A basis function is a polynomial factor that assigns a value or control point its influence in a Bézier formula. The curve combines these weighted contributions as the parameter t moves from 0 to 1. section 7.2.3
For a first-degree Bézier function, the basis polynomials are 1 − t and t: B(t) = (1 − t)y₁ + ty₂. At t = 0, the first factor selects y₁; at t = 1, the second selects y₂. Between them, the factors smoothly interpolate the two values. section 7.2.3
Interpolating two values
With y₁ = 0.2 and y₂ = 0.7, B(t) = (1 − t)·0.2 + t·0.7. At t = 0.4, the basis values are 0.6 and 0.4, producing 0.6·0.2 + 0.4·0.7 = 0.4. The basis values therefore act like changing blend weights. section 7.2.3
Higher-degree Bézier formulas use several basis polynomials:
- Quadratic curves use (1 − t)², 2(1 − t)t, and t² to combine three control-point coordinates. section 7.2.3
- Cubic curves use four basis terms, including (1 − t)³, 3(1 − t)²t, 3(1 − t)t², and t³. section 7.2.3
- NURBS scale basis polynomial functions; increasing their common degree trades fewer segments for greater continuity. section 7.3.1
