text:   Surfaces and Manifolds",4,"['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. \n\n<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.']",19,11,818