1 Definition and basic properties
1.1 Formal definition on \([0,1]\)
For a fixed nonnegative integer \(n\), the Bernstein basis on the unit interval \([0,1]\) is the collection of \(n+1\) polynomials \[ B_{i,n}(t)=\binom{n}{i}t^{i}(1-t)^{n-i},\quad i=0,1,\dots,n. \] Each basis element is a polynomial in \(t\) of degree \(n\), defined for all real \(t\) but especially interpreted on \([0,1]\) in approximation and geometric contexts.
1.2 Degree, indexing, and binomial coefficients
The index \(i\) selects the position within the family, with \(i=0\) giving the factor \((1-t)^n\) and \(i=n\) giving \(t^n\). The binomial coefficient \[ \binom{n}{i}=\frac{n!}{i!(n-i)!} \] controls how strongly each basis function weights the factors \(t^i\) and \((1-t)^{n-i}\). The basis therefore depends on two parameters: the degree \(n\) and the index \(i\).
1.3 Partition of unity
A central property is that the Bernstein basis forms a partition of unity on \([0,1]\): \[ \sum_{i=0}^{n} B_{i,n}(t)=1. \] This follows directly from the binomial theorem applied to \((t+(1-t))^n\). As a consequence, any Bernstein-form polynomial \[ p(t)=\sum_{i=0}^{n} c_i\, B_{i,n}(t) \] can be viewed as a weighted average of the coefficients \(c_i\) with nonnegative weights summing to one.
1.4 Nonnegativity and shape behavior
On the interval \([0,1]\), each factor \(t^{i}\) and \((1-t)^{n-i}\) is nonnegative, and binomial coefficients are nonnegative, so \[ B_{i,n}(t)\ge 0 \quad \text{for } t\in[0,1]. \] The basis functions typically have a single peak in \((0,1)\) and shift position as \(i\) changes. Increasing \(n\) produces increasingly “localized” basis functions that resemble probability mass concentrated around \(t\approx i/n\).
1.5 Endpoint values and boundary behavior
At the endpoints, \[ B_{0,n}(0)=1,\quad B_{n,n}(1)=1, \] and all other basis functions vanish at the respective endpoint (e.g., \(B_{i,n}(0)=0\) for \(i>0\)). For a Bernstein polynomial \(p(t)=\sum c_i B_{i,n}(t)\), this yields \[ p(0)=c_0,\qquad p(1)=c_n, \] which makes boundary values read directly from the first and last Bernstein coefficients.
2 Algebraic structure
2.1 Linear independence and spanning
The set \(\{B_{i,n}\}_{i=0}^{n}\) spans the vector space of polynomials of degree at most \(n\). In particular, it is linearly independent: if \[ \sum_{i=0}^{n} a_i B_{i,n}(t)=0\quad \text{for all } t, \] then all coefficients \(a_i\) must be zero. One way to see spanning is that each \(B_{i,n}\) has degree exactly \(n\) and the family has size \(n+1\), matching the dimension of the space.
2.2 Change of basis to monomials
Although Bernstein form is defined in terms of \(t\) and \((1-t)\), one can rewrite each basis polynomial as a linear combination of monomials \(t^k\). Expanding \((1-t)^{n-i}\) yields \[ B_{i,n}(t)=\binom{n}{i}\sum_{k=0}^{n-i} \binom{n-i}{k}(-1)^k\, t^{i+k}, \] so \(B_{i,n}\) becomes a signed mixture of monomials from degree \(i\) to \(n\). This provides a systematic but algebraically involved map between Bernstein coefficients \(\{c_i\}\) and standard polynomial coefficients \(\{a_k\}\).
2.3 Recurrence relations
Bernstein polynomials satisfy relations that connect neighboring indices and degrees. A common identity expresses \(B_{i,n}(t)\) in terms of basis functions of lower degree: \[ B_{i,n}(t)=(1-t)\,B_{i,n-1}(t)+t\,B_{i-1,n-1}(t), \] with appropriate conventions when indices fall outside \(0,\dots,n-1\). Such recurrences underpin both theoretical arguments and efficient algorithms.
2.4 Derivatives of Bernstein basis polynomials
The derivative of a Bernstein basis polynomial can be written as a scaled difference of Bernstein polynomials of lower degree: \[ \frac{d}{dt}B_{i,n}(t)=n\bigl(B_{i-1,n-1}(t)-B_{i,n-1}(t)\bigr), \] using the understanding that terms with out-of-range indices are treated as zero. For a Bernstein polynomial \(p(t)=\sum c_i B_{i,n}(t)\), this implies that derivatives correspond to coefficient differences, clarifying why control-point differences govern tangents and rates of change.
2.5 Integral identities and moments
Integrals of Bernstein polynomials admit closed forms involving beta functions. Specifically, \[ \int_{0}^{1} B_{i,n}(t)\,dt=\frac{1}{n+1}, \] which is consistent with symmetry over \(i\) and the partition of unity. More general “moments” of the form \(\int_0^1 t^m B_{i,n}(t)\,dt\) can also be expressed using combinatorial coefficients, reflecting that the Bernstein basis behaves like a family of beta densities (up to normalization).
3 Bernstein polynomials and geometric interpretation
3.1 Probabilistic viewpoint (binomial distribution)
The basis function \(B_{i,n}(t)\) matches the mass function of a binomial random variable. If \(X\sim \text{Bin}(n,t)\), then \[ \Pr(X=i)=\binom{n}{i}t^i(1-t)^{n-i}=B_{i,n}(t). \] This interpretation explains the localization near \(t\approx i/n\) and connects many properties to standard probabilistic identities, such as expectation and variance (with the caveat that \(t\) is a parameter rather than a random outcome).
3.2 Control-point interpretation for polynomials
When one writes \[ p(t)=\sum_{i=0}^{n} c_i\, B_{i,n}(t), \] the coefficients \(c_i\) function like control points. Because the basis forms a partition of unity and is nonnegative on \([0,1]\), the value \(p(t)\) is a convex combination of the control values \(\{c_i\}\) at each parameter \(t\).
3.3 Convex hull property
From the convex-combination viewpoint, if \(c_i\) are real numbers, then for \(t\in[0,1]\), \[ \min_i c_i \le p(t)\le \max_i c_i. \] In geometric modeling with vector-valued control points \(\{P_i\}\), the curve point \(C(t)\) lies in the convex hull of \(\{P_i\}\) for all \(t\). This property is a key reason Bernstein-based constructions are popular in computer graphics.
3.4 Variation-diminishing behavior (intuition)
Bernstein representations exhibit “variation-diminishing” tendencies: as one maps a control sequence through Bernstein blending, the number of sign changes (or oscillations) in the resulting function cannot increase. Intuitively, the basis acts as a smoothing kernel because each \(B_{i,n}\) is nonnegative and concentrated around a specific parameter region. This helps explain why Bernstein forms often yield visually stable curves under moderate perturbations.
4 Bézier curves and Bernstein basis
4.1 Polynomial representation of Bézier curves
A Bézier curve of degree \(n\) can be defined using Bernstein basis polynomials: \[ C(t)=\sum_{i=0}^{n} P_i\, B_{i,n}(t),\quad t\in[0,1], \] where \(P_0,\dots,P_n\) are control points (typically vectors in \(\mathbb{R}^d\)). The curve is therefore a polynomial parametrization whose coefficients are geometrically interpretable through the basis weights.
4.2 Control points and evaluation
To evaluate \(C(t)\), one computes the basis values \(B_{i,n}(t)\) and forms the weighted sum of control points \(P_i\). Since the basis weights are nonnegative and sum to 1 on \([0,1]\), the computation corresponds to interpolating within the convex hull of the control polygon. The basis’s endpoint behavior ensures \[ C(0)=P_0,\qquad C(1)=P_n, \] so the curve starts and ends at the first and last control points.
4.3 Endpoint tangents and derivatives
Derivatives of Bézier curves are closely related to differences between neighboring control points. Using the derivative identity for Bernstein basis functions, one obtains that the tangent direction at \(t=0\) depends on \(P_1-P_0\), and the tangent at \(t=1\) depends on \(P_n-P_{n-1}\). Higher derivatives relate to higher-order finite differences, which is why control-point spacing influences curvature near the ends.
4.4 Subdivision and de Casteljau’s algorithm
Bézier curves can be subdivided into curves on subintervals using de Casteljau’s algorithm. This method recursively forms linear interpolations between control points, effectively performing the evaluation in a numerically robust manner. Rather than computing powers of \(t\) and \((1-t)\) directly, it relies on repeated convex combinations, aligning with the Bernstein convex-hull structure. Subdivision yields new control points for the left and right curve segments while preserving the geometry.
5 Operations in Bernstein form
5.1 Addition and scalar multiplication in the basis
If two polynomials are expressed in Bernstein form of the same degree, \[ p(t)=\sum_{i=0}^n a_i B_{i,n}(t),\qquad q(t)=\sum_{i=0}^n b_i B_{i,n}(t), \] then their sum and scalar multiple are straightforward: \[ p(t)+q(t)=\sum_{i=0}^n (a_i+b_i)B_{i,n}(t),\qquad \alpha p(t)=\sum_{i=0}^n (\alpha a_i)B_{i,n}(t). \] This linearity is one reason Bernstein representations are convenient for constructive geometry and design workflows.
5.2 Multiplication and degree elevation
The product of two Bernstein polynomials of degrees \(n\) and \(m\) is generally expressible in a Bernstein basis of degree \(n+m\). Practically, multiplication requires conversion because the degree changes under multiplication of polynomials. This phenomenon motivates degree elevation as a general technique for aligning representations before performing operations.
5.3 Degree elevation formulas
Degree elevation rewrites a polynomial in Bernstein form of degree \(n\) as one of degree \(n+1\) without changing the underlying function. If \[ p(t)=\sum_{i=0}^{n} c_i B_{i,n}(t), \] then there exist coefficients \(c'_i\) for \(i=0,\dots,n+1\) such that \[ p(t)=\sum_{i=0}^{n+1} c'_i B_{i,n+1}(t). \] The elevated coefficients can be chosen so that each new control point is a weighted combination of neighboring old control points, with weights determined by binomial coefficients. This preserves endpoint values while refining the control structure.
5.4 Conversion between different degrees
More generally, converting between Bernstein representations of different degrees is accomplished by repeated degree elevation (or, in some contexts, degree reduction when possible). This is important when combining curves, comparing control polygons, or setting up algorithms that require matching degrees across operands.
5.5 Linear system forms for interpolation
Given data values and a choice of parameter points \(\{t_j\}\), one can determine Bernstein coefficients by solving a linear system: \[ p(t_j)=\sum_{i=0}^n c_i B_{i,n}(t_j)=y_j,\quad j=0,\dots,N. \] When \(N=n\) and the system is nonsingular, the coefficients are uniquely determined. Using Bernstein form can improve interpretability: coefficients correspond to control points, and constraints at \(t=0\) and \(t=1\) directly fix \(c_0\) and \(c_n\).
6 Approximation and interpolation theory
6.1 Bernstein approximation theorem (statement-level overview)
The Bernstein approximation theorem states that for suitable continuous functions \(f\) on \([0,1]\), one can approximate \(f\) uniformly by Bernstein polynomials constructed from sampled values of \(f\) at a grid of points: \[ B_n(f)(t)=\sum_{i=0}^{n} f\!\left(\frac{i}{n}\right) B_{i,n}(t). \] As \(n\to\infty\), \(B_n(f)\) converges to \(f\) uniformly on \([0,1]\). This provides a constructive approximation scheme tied directly to the basis.
6.2 Convergence and smoothness considerations
While uniform convergence holds for continuous functions, the rate of convergence depends on the function’s regularity. For smoother functions, error decays more rapidly, and derivatives of the approximants can converge under stronger assumptions. The basis’s probabilistic interpretation connects approximation quality to how averaging with binomial weights smooths local behavior.
6.3 Relation to other polynomial bases
Bernstein polynomials form one of several classical polynomial bases used in approximation theory (others include power basis, Lagrange basis, and orthogonal polynomial bases). Unlike orthogonal bases, Bernstein polynomials are not defined by inner-product orthogonality; instead, their strength lies in nonnegativity, endpoint structure, and geometric convexity. Converting between bases is possible but may change numerical properties.
6.4 Error behavior and qualitative estimates
Error analysis often compares \(f(t)\) to an averaged version of \(f\) around \(t\). Because \(B_{i,n}(t)\) behaves like binomial weights centered near \(i/n\), Bernstein approximation can be understood as a controlled smoothing operator. Qualitative bounds typically relate the approximation error to moduli of continuity or to derivatives through inequalities that quantify how the smoothing scale shrinks as \(n\) grows.
6.5 Applications to smoothing and fitting
Bernstein approximation naturally supports smoothing: the operator \(f \mapsto B_n(f)\) tends to damp high-frequency features. This can be useful in data fitting and in constructing polynomial surrogates that preserve key constraints such as boundary values or monotonic trends when combined with additional techniques. In practice, Bernstein form also supports shape-preserving design strategies because of its convex hull and variation-diminishing tendencies.
7 Multivariate extension
7.1 Bernstein basis on triangles (outline)
Multivariate Bernstein bases extend the one-dimensional construction to domains such as triangles using barycentric coordinates. For a triangular region with barycentric coordinates \((u,v,w)\) satisfying \(u+v+w=1\), a typical Bernstein polynomial of total degree \(n\) takes the form \[ B_{i,j,k}^{(n)}(u,v,w)=\frac{n!}{i!\,j!\,k!}u^{i}v^{j}w^{k}, \quad i+j+k=n. \] These functions are nonnegative and sum to 1 over the simplex, preserving the partition-of-unity and convex-hull interpretations.
7.2 Bernstein basis on tensor-product domains
On rectangular or box-like domains, one can form multivariate bases by tensor products of the univariate Bernstein basis. For example, on \([0,1]^2\), \[ B_{i,n}(t)\,B_{j,m}(s) \] provides a basis element of bidegree \((n,m)\). Tensor-product formulations align well with grid-based control structures and allow separate handling of each parameter direction.
7.3 Bézier patches and surfaces
Bézier surfaces generalize Bézier curves by using multivariate Bernstein bases: \[ S(u,v)=\sum_{i=0}^{n}\sum_{j=0}^{m} P_{i,j}\, B_{i,n}(u)\,B_{j,m}(v), \] where \(P_{i,j}\) are control points arranged in a control net. The resulting surface inherits convex-hull-like behavior: for each fixed \((u,v)\), the surface point is a convex combination of the control points weighted by products of Bernstein basis functions.
7.4 Key properties preserved under extension
Many core properties extend from the univariate setting: nonnegativity on the domain, partition of unity, endpoint interpolation in appropriate parameter directions, and convexity-based geometric constraints. While multivariate shape analysis can be more complex, the Bernstein framework remains a common foundation in surface modeling due to these inherited structural features.
8 Computational aspects
8.1 Numerical stability considerations
Direct evaluation of \(B_{i,n}(t)\) via powers can lead to numerical issues for large \(n\) due to underflow/overflow and cancellation effects. Bernstein-based computations are often stabilized either by using recursion for basis values or by relying on de Casteljau’s algorithm, which restructures computations into sequences of convex combinations. These approaches tend to be more robust in floating-point arithmetic.
8.2 Efficient evaluation strategies
Efficient strategies include computing all basis values for a given \(t\) using recurrence relations, or evaluating Bézier curves directly through de Casteljau’s algorithm. In contexts requiring repeated evaluations for many \(t\) values, precomputing or caching basis quantities can reduce overhead.
8.3 Using de Casteljau vs. direct evaluation
- Direct evaluation: computes \(\sum P_i B_{i,n}(t)\) from basis values. It can be efficient for small degrees but may degrade for high degrees.
- de Casteljau: evaluates via recursive linear interpolation among control points. It is widely used because it avoids large intermediate powers and maintains geometric invariants implied by convex combinations.
8.4 Complexity and implementation notes
de Casteljau’s algorithm for degree \(n\) uses on the order of \(n^2\) interpolation steps per evaluation point. Direct evaluation also requires summing \(n+1\) terms, but the cost depends on how basis values are computed. Practical implementations balance these costs against numerical reliability, especially in interactive graphics where degrees are moderate and evaluations are frequent.
8.5 Practical pitfalls (scaling and degree growth)
As \(n\) grows, basis functions become increasingly peaked and can be challenging to represent accurately. Additionally, operations such as multiplication may raise degree, potentially increasing computational expense and amplifying numerical sensitivity. Implementers often mitigate these issues by degree management (keeping degrees controlled), using stable evaluation schemes, and carefully scaling parameters to maintain floating-point accuracy.