DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Piecewise Bernstein representation

Definition

Let $s \geq 1$ be an integer and let $a = x_{0} < \dots < x_{s} = b$ be a finite partition, and on each cell $[x_{i}, x_{i + 1}]$ choose an integer degree $n_{i} \geq 0$ and real controls $c_{i, 0}, \dots, c_{i, n_{i}}$ . For $t = (x - x_{i}) / (x_{i + 1} - x_{i}) \in [0, 1]$ , define the Bernstein basis polynomial $B_{k, n_{i}} (t) = ι (\binom{n_{i}}{k}) t^{k} (1 - t)^{n_{i} - k}, 0 \leq k \leq n_{i},$ where $(\binom{n_{i}}{k})$ is the natural binomial coefficient from The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $(\binom{n}{k}) : = ∣ [n]^{k} ∣$ , $ι$ is the canonical embedding of The canonical natural $ι (n) = n \cdot 1_{F}$ of a field into $R$ , and finite sums use Finite sums and finite products, by recursion. The cell polynomial is $p_{i} (x) = \sum_{k = 0}^{n_{i}} c_{i, k} B_{k, n_{i}} (t)$ . These cell polynomials form a piecewise Bernstein representation. To require continuity at a shared node, impose the additional endpoint-control condition $c_{i, n_{i}} = c_{i + 1, 0}$ ; continuity is not part of the representation unless that condition is imposed.

For degree two this formula reads $p_{i} (x) = c_{i, 0} (1 - t)^{2} + ι (\binom{2}{1}) c_{i, 1} t (1 - t) + c_{i, 2} t^{2}$ .

Depends on

Used by

Dependency tree · two levels

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Sources

Rida T. Farouki, The Bernstein polynomial basis: a centennial retrospective, Computer Aided Geometric Design 29 (2012), pp. 379–419 (standard reference, not scraped)