• q ≥ p = 1 • p = q = 2 Proof. The p = 1 case is trivial as all 1-simplices have the same regularity so we have ϑ$_{K}$ = 8 and can take M$_{ϵ}$ = M ′ ϵ = K . For the p = q = 2 case, we proceed in two steps. First we will superimpose a square grid on K (orienting it to bound the minimum angle created between its edges and those of K ), creating a cell complex which is a refinement of K . Next we use Shewchuk’s Terminator algorithm [38] to further refine the cell complex back into a simplicial complex with bounds on the minimum angle and, crucially, restrictions on where these small angles can be so that we can obtain regularity bounds. By superimposing a fine enough square grid, we can force the small angles (whether already present in the complex or newly created) to occur only in a small measure subset of the complex. Pick δ > 0 small enough that the set has measure less than ϵ . Let G be a finite square grid in R 2 whose cells each have diameter δ such that G covers the underlying space of K in any rotation. Note that there are only two directions present in G so if we bound all possible angles created between these directions and the edges of K , we can bound the minimum new angle created by superimposing G . Further let E θ = { ψ ∈ [0 , 2 π ) | | φ − ψ | < θ for some φ ∈ E } . This is the set of angles to 68 Let w ∈ R 2 be a fixed unit vector and define E =$^{{}$ φ, φ + π 2 | φ is the angle between u − v and w for some edge ( u, v ) ∈ K$^{}}$ . { x ∈ R 2 | y lies on the 1-skeleton of K , ‖ x − y ‖ < 3 δ }