avoid when rotating G in order to guarantee all created angles will be θ or larger. Denote by N < ∞ the cardinality of E and note that [0 , 2 π ) \ E π 2 N has positive measure so there exist rotations of the square grid that create no new angles smaller than π 2 $_{N}$. After superimposing a suitably rotated version of G , we obtain a new cellular complex which is a refinement of K . This is a planar straight line graph which can be used as input to Shewchuk’s Terminator algorithm [38] which refines it into a simplicial complex M$_{ϵ}$ with guarantees about the minimum angle bound of the resulting complex and where the small angles can occur. In particular, if θ be the minimum angle in the cellular complex (either present originally or added in the square grid superposition), then the minimum angle of M$_{ϵ}$ $_{is at least arcsin((}$√$_{3}$$_{/}$$_{2) sin(}$$_{θ/}$$_{2)). Furthermore, no angles less than 30}$ ◦ are created by the algorithm except in the vicinity of angles less than 60 $^{◦}$. Specifically, newly created small angles must be part of a skinny triangle whose circumcenter encroaches upon a subsegment cluster bearing a small input angle. As all such subsegment clusters must be contained within a distance of 2 δ of the 1-skeleton of K , we have that all small angles in M$_{ϵ}$ are within 3 δ of the 1-skeleton of K . Let M ′ ϵ be the subcomplex of M$_{ϵ}$ containing all triangles not fully contained in the 3 δ tube, noting that all angles in M ′ ϵ are at least 30 ◦ so by Lemma 3.3.3 we have We may take β to be this quantity, noting that it is independent of ϵ and K . The mini- mum angle bound θ for M$_{ϵ}$ and Lemma 3.3.3 give us a bound α$_{K}$ for ϑ$_{M}$$_{ϵ}$ (independent of ϵ ). The following theorem shows that the bounds in Theorem 3.3.1 may be replaced 69 ϑ$_{M}$ ′ ϵ ≤ 48 π cot(15 $^{◦}$) 2 + 4 cot(15 $^{◦}$) = $_{4(2 +}$√$_{3)(24 + 12}$ √$_{3 +}$$_{π}$$_{)}$ π .