## 7 Proofs of lemmas in Section 2  
For proving Lemma 2 we need some standard results from sampling theory [20; 21]. Recall that we have assumed each \(2\)-face in the domain to be a compact subset of a smooth (\(C^{2}\)) \(2\)-manifold without boundary. This helps us to carry forward a result on normal variation from smooth surfaces straightforwardly to the surface patches with boundaries. If surface elements are not assumed to be a subset of smooth surfaces without boundary, normal variation proof becomes more complicated with a worse error bound. Let \(\sigma\) be any \(2\)-face which by assumption is a subset of a smooth \(2\)-manifold \(\sigma^{*}\) without boundary. The surface normal \(\mathbf{n}_{x}\) to \(\sigma\) at \(x\) is the surface normal of \(\sigma^{*}\) at \(x\). Let \(M\) denote the medial axis of \(\sigma^{*}\)that is, \(M\) is the closure of all points in \(\mathbb{R}^{3}\) which have two or more closest points in \(\sigma^{*}\). Borrowing from [20], we define the local feature size \(\mathrm{lfs}_{x}=d(x,M)\). In the analysis, we use a global lower bound on the \(\mathrm{lfs}\) function. We define:  
\[\rho_{\sigma}=\inf_{x\in\sigma}\mathrm{lfs}_{x}.\]  
Recall that \(\hat{d}(x,y)\) denotes the weighted distance between two weighted points \(x\) and \(y\), that is, \(\hat{d}(x,y)=\sqrt{d(x,y)^{2}-\omega_{x}^{2}-\omega_{y}^{2}}\) where \(\omega_{x}\) and \(\omega_{y}\) are the weights of \(x\) and \(y\) respectively. For a triangle \(t\), consider the disk whose weighted distance to all of \(t\)'s vertices (possibly weighted) is zero. The radius of this disk is called the _orthoradius_ of \(t\). **Lemma 10**: _Let \(\varepsilon\in(0,1/3)\) be some constant._  
* _For any two points_ \(x\) _and_ \(y\) _in_ \(\sigma\) _such that_ \(d(x,y)\leqslant\varepsilon\rho_{\sigma}\)_,_
* _the angle between the surface normals_ \(\mathbf{n}_{x}\) _at_ \(x\) _and_ \(\mathbf{n}_{y}\) _at_ \(y\) _is at most_ \(2\varepsilon\)_;_
* _the angle between_ \(xy\) _and the surface normal at_ \(x\) _is at least_ \(arccos(\varepsilon/2)\)_._
* _Let_ \(pqr\) _be a triangle with vertices on_ \(\sigma\) _which are possibly weighted with a weight_ \(\omega\rho_{\sigma}\) _for_ \(\omega<1\)_. Furthermore, let the orthoradius of_ \(pqr\) _be no more than_ \(\varepsilon\rho_{\sigma}\)_. Then,_ \(\angle_{a}(\mathbf{n}_{pqr},\mathbf{n}_{p})\leqslant 5\sqrt{\varepsilon^{2}+ \omega^{2}}\)_._  
(i.a) Follows from the normal variation result in [22]. (i.b) Follows from the Edge Normal Lemma in [21]. (ii) Let \(R\) and \(\hat{R}\) be the circumradius and orthoradius of \(pqr\) respectively. For a sufficiently small \(\omega\), the circumcenter \(x\) lies inside the orthocircle of \(pqr\). Then, minimum weighted distance of \(x\) to any of three vertices of \(pqr\), is a lower bound on \(\hat{R}\), that is,  
\[R^{2}-\omega^{2}\rho_{\sigma}^{2} \leqslant \hat{R}^{2}\] \[R^{2} \leqslant \varepsilon^{2}\rho_{\sigma}^{2}+\omega^{2}\rho_{\sigma}^{2}\] \[\leqslant (\varepsilon^{2}+\omega^{2})\rho_{\sigma}^{2}.\]  
Without loss of generality, assume that \(p\) subtends the largest angle in \(pqr\). Plugging \(R_{pqr}=R\leqslant\sqrt{\varepsilon^{2}+\omega^{2}}\rho_{\sigma}\) and using \(\mathrm{lfs}_{(}p)\geqslant\rho_{\sigma}\) in the Triangle Normal Lemma (Lemma 3.5 in [21]), we get that  
\[\angle_{a}(\mathbf{n}_{p},\mathbf{n}_{pqr}) \leqslant \arcsin\sqrt{\varepsilon^{2}+\omega^{2}}+\arcsin\left(\frac{2}{ \sqrt{3}}\left(2\arcsin\sqrt{\varepsilon^{2}+\omega^{2}}\right)\right)\] \[\leqslant 5\sqrt{\varepsilon^{2}+\omega^{2}}.\]  
Consider a weighted vertex \(p\in S\) on a \(2\)-face \(\sigma\). For any facet \(F\) of \(V_{p}\), we use \(H_{F}\) to denote the plane of \(F\). Let \(\lambda\leqslant\frac{1}{32}\rho_{\sigma}\). Let \(B=B(p,2\lambda)\). By the Feature Ball Lemma in [21]\(B\cap\sigma\) is a 2-disk.