TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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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.
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A planar triangulation has linearly many triangles and edges

Statement

For a triangulation of the convex hull of $n$ distinct planar sites with affine dimension two, let $k$ be the number of sites on the hull boundary, $T$ the number of triangles, and $E$ the number of edges. Then $T = 2 n - 2 - k, E = 3 n - 3 - k .$ In particular $T < 2 n$ and $E < 3 n$ .

Facts & Assumptions

Given: A finite triangulation of the convex hull of $n$ distinct sites of affine dimension two.

[F1]

Planar triangulations, Delaunay structures, and Voronoi cells: every hull-boundary site is a vertex, the boundary is subdivided at all such sites, and bounded faces are exactly the chosen triangles.

[F2]

Euler counting for a finite planar triangulation: $T = 2 n - 2 - k$ and $E = 3 n - 3 - k$ for this triangulation.

[A1]

The convex hull boundary is a polygon with exactly $k$ boundary edges after subdivision at its $k$ sites.

Proof

1.1givenF1F2A1

By [F1] and [A1], the triangulation has $k$ hull boundary sites and therefore exactly $k$ subdivided hull edges. Then [F2] gives $T = 2 n - 2 - k$ .

2.1F2step 1.1algebra∎

By [F2], $E = 3 n - 3 - k$ . Since $k \geq 3$ , the formulas imply $T < 2 n$ and $E < 3 n$ .

Depends on

Used by

Dependency tree · two levels

5 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 13 (standard reference, not scraped)