ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generated

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.

Affine maps for a triangle and a parallelogram

Example

The orientation-preserving map $F (x, y) = (2 x, 3 y)$ sends the reference closed triangle $\overline{{\hat{S}}_{2}} = conv {(0, 0), (1, 0), (0, 1)}$ to the triangle with vertices $(0, 0), (2, 0), (0, 3)$ and area $3$ . The affine map $G (s, t) = (2 s + t, 3 t)$ sends the unit square to the parallelogram with vertices $(0, 0), (2, 0), (1, 3), (3, 3)$ and area $6$ .

Facts & Assumptions

Given: The reference simplex and the convention that affine maps use their constant Jacobian matrix, as in Matching affine simplicial meshes and shape regularity.

[F1]

The reference triangle has vertices $(0, 0), (1, 0), (0, 1)$ , and a reference map is affine with a constant matrix. (Matching affine simplicial meshes and shape regularity)

Verification

1.1F1givenalgebra

The first matrix is $A = (\begin{array}{cc} 2 & 0 \\ 0 & 3 \end{array})$ , with determinant $6 > 0$ . Applying it to the three vertices gives $(0, 0), (2, 0), (0, 3)$ . This is a right triangle with legs $2$ and $3$ , so its area is $\frac{1}{2} \cdot 2 \cdot 3 = 3$ .

2.1F1step 1.1algebra∎

The second matrix is $B = (\begin{array}{cc} 2 & 1 \\ 0 & 3 \end{array})$ , with determinant $6 > 0$ . Applying it to the square's vertices gives $(0, 0), (2, 0), (1, 3), (3, 3)$ , whose opposite edges are parallel because they are translates of the two column vectors $(2, 0)$ and $(1, 3)$ . Hence the image is a parallelogram with base length $2$ and altitude $3$ , so its area is $2 \cdot 3 = 6$ . It is an affine image of a square, not a simplicial cell.

Depends on

Matching affine simplicial meshes and shape regularity

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

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Sources

Jean-Luc Guermond, MATH 661 Fall 2025, Chapter 8 (standard reference, not scraped)
Luca Heltai, Numerical Methods for Partial Differential Equations, Lecture 2 (standard reference, not scraped)