ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck pass

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.

A scattered affine interpolant on one Delaunay cell

Statement

On the Delaunay triangle with vertices $(0, 0), (2, 0), (0, 2)$ and vertex data $0, 2, 2$ , the query $(1 / 2, 1 / 2)$ has barycentric weights $1 / 2, 1 / 4, 1 / 4$ and affine interpolated value $1$ .

Facts & Assumptions

Given: The three indexed dyadic sites $(0, 0), (2, 0), (0, 2)$ with respective scalar data $0, 2, 2$ , and query $x = (1 / 2, 1 / 2)$ .

[F1]

Finite indexed geometric site sets and duplicate handling: these are distinct exact dyadic sites.

[F2]

Planar triangulations, Delaunay structures, and Voronoi cells: a triangulation of three noncollinear sites has their convex hull as its single triangular cell.

[F3]

Barycentric point location, affine interpolation, and nearest-site queries: barycentric coordinates sum to one, reconstruct the point, and the interpolant is their weighted data sum.

[F4]

Barycentric interpolation and nearest-site queries are correct: the coordinates determine exact membership and the unique affine interpolant.

[F5]

Exact planar insertion terminates with a weak empty-circle triangulation: the cell is weakly empty-circle and is returned by the algorithm.

[A1]

No other sites are included in this example's site set.

Proof

1.1givenF1F3algebra

Let $λ_{0}, λ_{1}, λ_{2}$ correspond to $(0, 0), (2, 0), (0, 2)$ . The coordinate equations give $2 λ_{1} = 1 / 2$ and $2 λ_{2} = 1 / 2$ , hence $λ_{1} = λ_{2} = 1 / 4$ ; their sum constraint gives $λ_{0} = 1 / 2$ . All three are nonnegative, so the query lies in the triangle by [F3].

2.1F3F4step 1.1algebra

The interpolated value is $0 \cdot λ_{0} + 2 \cdot λ_{1} + 2 \cdot λ_{2} = 2 (1 / 4) + 2 (1 / 4) = 1$ . Uniqueness follows from [F4].

3.1F2F5A1algebra∎

The three sites are noncollinear, so by [F2] their convex hull is their single triangle. Its circumcircle has center $(1, 1)$ and radius squared $2$ ; there are no other sites by [A1], so it is weakly empty, and [F5] returns this cell.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Mark de Berg et al., Computational Geometry: Algorithms and Applications, 3rd ed., Chapter 9 (standard reference, not scraped)
David M. Mount, CMSC 754 Computational Geometry, Lecture 11 (standard reference, not scraped)