Orientation Preserving Maps of the Square Grid II

Abstract

AbstractFor a finite set $$S\subset {\mathbb {R}}^2$$ S ⊂ R 2 , a map $$\varphi :S\rightarrow {\mathbb {R}}^2$$ φ : S → R 2 is orientation preserving if for every non-collinear triple $$u,v,w\in S$$ u , v , w ∈ S the orientation of the triangle u, v, w is the same as that of the triangle $$\varphi (u),\varphi (v),\varphi (w)$$ φ ( u ) , φ ( v ) , φ ( w ) . Assuming that $$\varphi :G_n\rightarrow {\mathbb {R}}^2$$ φ : G n → R 2 is an orientation preserving map where $$G_n$$ G n is the grid $$\{0,\pm 1,\dots ,\pm n\}^2$$ { 0 , ± 1 , ⋯ , ± n } 2 and n is large enough we prove that there is a projective transformation $$\mu :{\mathbb {R}}^2\rightarrow {\mathbb {R}}^2$$ μ : R 2 → R 2 such that $$\Vert \mu \circ \varphi (z)-z\Vert =O(1/n)$$ ‖ μ ∘ φ ( z ) - z ‖ = O ( 1 / n ) for every $$z\in G_n$$ z ∈ G n .