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Opposite orientations cancel on a two-triangle mesh

Statement

Split the unit square along the diagonal from v0=(0,0) to v2=(1,1), with v1=(1,0) and v3=(0,1). Orient the two triangles counterclockwise as [v0,v1,v2] and [v0,v2,v3]. Their shared diagonal occurs once with each orientation and cancels in the sum of their oriented boundaries.

Facts & Assumptions

Given: The alternating simplicial boundary convention of Oriented simplicial cell chains and boundary and the matching two-cell mesh.

[F1]

For an ordered triangle, ∂[v0,v1,v2]=[v1,v2]−[v0,v2]+[v0,v1]. (Oriented simplicial cell chains and boundary)

[F2]

A shared facet of a matching mesh has opposite induced orientations in the two cell boundaries. (Interior facet orientations cancel in a matching mesh)

Proof

1.1F1givenalgebra

Apply [F1] to the first triangle and the ordered triangle [v0,v2,v3] to obtain ∂[v0,v1,v2]=[v1,v2]−[v0,v2]+[v0,v1], ∂[v0,v2,v3]=[v2,v3]−[v0,v3]+[v0,v2]. The edge [v0,v2] appears with coefficient −1 in the first boundary and +1 in the second.

2.1F1F2step 1.1algebra∎

Adding the two formulas cancels the shared diagonal and leaves [v0,v1]+[v1,v2]+[v2,v3]−[v0,v3], which is the oriented outer boundary since −[v0,v3]=[v3,v0]. This explicit calculation agrees with the general cancellation rule [F2].

Depends on

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