Bernstein polynomials: B_{k,n}(t)=C(n,k)t^k(1-t)^{n-k}, k=0,...,n. Key properties: non-negative on [0,1], partition of unity (sum=1), and form a basis. Bézier curves: B(t)=Σ P_k·B_{k,n}(t). Weierstrass approximation theorem proof uses Bernstein polynomials.
How to Use
Enter n and t:
- Values: All B_{k,n}(t)
- Verify: Sum = 1
- Bézier: Curve evaluation
Bézier Curves
Control points P_0,...,P_n define a Bézier curve: B(t)=Σ P_k·B_{k,n}(t). Degree 3 (cubic) is most common: B(t)=(1-t)³P₀+3t(1-t)²P₁+3t²(1-t)P₂+t³P₃. Used in PostScript, TrueType fonts, SVG paths.
Special Properties
Non-negativity: B_{k,n}≥0 on [0,1]. Partition of unity: Σ_k B_{k,n}=1. Symmetry: B_{k,n}(t)=B_{n-k,n}(1-t). Maximum at t=k/n with value C(n,k)(k/n)^k((n-k)/n)^{n-k}.
Step-by-Step Instructions
- 1Enter degree n.
- 2Enter parameter t.
- 3Compute all B_{k,n}(t).
- 4Verify sum = 1.
- 5See Bézier link.