text: 436,57,1193967646477127680,548718054,Marc Khoury,"How good is a triangulation as an approximation to a smooth surface? In a new paper, we prove sharp bounds on the interpolation and normal error for points clouds sampled from smooth surfaces and manifolds. With Jonathan Shewchuk.  <LINK> @paul_pearce Depends on the triangulation and the density of the samples. The bounds are statements of the type if a triangle is ""small"", wrt a certain measure, and the sample is dense, then the triangle normal closely approximates the true normal at the vertices. @paul_pearce Given a triangulation these statements allow you to measure its quality. A lot of the effort is in establish the sharpest possible bound wrt constants. Many of these results are central to provable manifold reconstruction and haven't been improved in years. @paul_pearce Even better: arbitrarily good.",https://arxiv.org/abs/1911.03424,"How good is a triangulation as an approximation of a smooth curved surface or manifold? We provide bounds on the {\em interpolation error}, the error in the position of the surface, and the {\em normal error}, the error in the normal vectors of the surface, as approximated by a piecewise linearly triangulated surface whose vertices lie on the original, smooth surface. The interpolation error is the distance from an arbitrary point on the triangulation to the nearest point on the original, smooth manifold, or vice versa. The normal error is the angle separating the vector (or space) normal to a triangle from the vector (or space) normal to the smooth manifold (measured at a suitable point near the triangle). We also study the {\em normal variation}, the angle separating the normal vectors (or normal spaces) at two different points on a smooth manifold. Our bounds apply to manifolds of any dimension embedded in Euclidean spaces of any dimension, and our interpolation error bounds apply to simplices of any dimension, although our normal error bounds apply only to triangles. These bounds are expressed in terms of the sizes of suitable medial balls (the {\em empty ball size} or {\em local feature size} measured at certain points on the manifold), and have applications in Delaunay triangulation-based algorithms for provably good surface reconstruction and provably good mesh generation. Our bounds have better constants than the prior bounds we know of---and for several results in higher dimensions, our bounds are the first to give explicit constants. ","Approximation Bounds for Interpolation and Normals on Triangulated