Bernstein Polynomial Calculator
Interactive Bernstein Polynomial & Bézier Curve Generator
Real-time visualization of Bernstein basis polynomials — the mathematical foundation of Bézier curves
About the Bernstein Polynomial Calculator
The Bernstein Polynomial Calculator is a scientifically accurate, interactive tool that computes and visualizes Bernstein basis polynomials — the mathematical foundation of Bézier curves and surfaces. Developed by Sergei Natanovich Bernstein in 1912 as part of his constructive proof of the Weierstrass approximation theorem, these polynomials are defined as Bᵢⁿ(t) = (n choose i) tⁱ (1−t)ⁿ⁻ⁱ where t ∈ [0,1]. This calculator implements the exact binomial coefficient formula with real-time plotting, making it an essential educational and design tool for students, mathematicians, animators, and computer graphics professionals. Built with pure HTML5 Canvas and JavaScript, it requires no external libraries and works instantly in any modern browser.
Importance of the Bernstein Polynomial Calculator
Bernstein polynomials are fundamental to modern computer graphics and approximation theory. They form the basis functions for Bézier curves — the industry-standard method for representing smooth curves in vector graphics, font design, animation, and CAD software. The Bernstein Polynomial Calculator allows users to explore how each basis polynomial contributes to the final Bézier curve, providing deep insight into curve behavior and control point influence. This direct visualization of the underlying mathematics is invaluable for understanding why Bézier curves are convex, variation-diminishing, and partition of unity — properties that make them ideal for design work. The mathematics follows the exact formulation documented in the authoritative Bernstein Polynomial article on Wikipedia.
User Guidelines
To use the Bernstein Polynomial Calculator effectively:
- Select Degree (n): Choose the polynomial degree (1 to 5). Higher degrees create more complex curves.
- Select Index (k): Choose which basis polynomial Bₖⁿ(t) to highlight (0 to n).
- Adjust Control Points: Move the control points that define the Bézier curve.
- Set t Value: The vertical red line shows the current parameter t and its contribution from each polynomial.
- Click "Calculate": Watch the Bernstein basis polynomials and resulting Bézier curve update instantly.
When and Why You Should Use the Bernstein Polynomial Calculator
Use this tool when you need to:
- Understand how control points influence Bézier curves
- Teach or learn the mathematical foundation of computer graphics
- Debug animation timing functions and easing curves
- Study approximation theory and polynomial interpolation
- Design custom curves with precise mathematical control
- Visualize the partition of unity and convex hull properties
Purpose of the Bernstein Polynomial Calculator
The primary purpose is to bridge the gap between abstract mathematics and practical computer graphics by providing an interactive visualization of Bernstein basis polynomials in action. By showing exactly how each polynomial contributes to the final curve at any parameter t, users gain profound insight into one of the most important concepts in computational geometry.
Scientific and Mathematical Foundation
The Bernstein basis polynomial of degree n is defined as:
where t ∈ [0,1], and i = 0,1,...,n
A Bézier curve is then formed as the weighted sum: B(t) = Σ Bᵢⁿ(t) Pᵢ
Key properties include partition of unity (Σ Bᵢⁿ(t) = 1), non-negativity, and symmetry.
Real-World Applications
Bernstein polynomials are used in:
- Bézier curves and surfaces in all vector graphics software
- Font design (TrueType, OpenType)
- CSS animation timing functions
- CAD/CAM systems and 3D modeling
- Statistical modeling and probability density estimation
Conclusion
The Bernstein Polynomial Calculator is an essential educational and professional tool for anyone working with curves in digital design or mathematics. By making the invisible mathematics behind Bézier curves visible and interactive, it transforms abstract theory into tangible understanding. Whether you're a student learning approximation theory, a designer perfecting smooth paths, or a developer implementing animation systems, this calculator provides the insight needed for mastery. Explore more advanced mathematics tools at Agri Care Hub and deepen your knowledge of the foundational Bernstein Polynomial.