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How good is a triangulation as an approximation of a smooth curved surface ormanifold? We provide bounds on the {\em interpolation error}, the error in theposition of the surface, and the {\em normal error}, the error in the normalvectors of the surface, as approximated by a piecewise linearly triangulatedsurface whose vertices lie on the original, smooth surface. The interpolationerror is the distance from an arbitrary point on the triangulation to thenearest point on the original, smooth manifold, or vice versa. The normal erroris the angle separating the vector (or space) normal to a triangle from thevector (or space) normal to the smooth manifold (measured at a suitable pointnear the triangle). We also study the {\em normal variation}, the angleseparating the normal vectors (or normal spaces) at two different points on asmooth manifold. Our bounds apply to manifolds of any dimension embedded inEuclidean spaces of any dimension, and our interpolation error bounds apply tosimplices of any dimension, although our normal error bounds apply only totriangles. These bounds are expressed in terms of the sizes of suitable medialballs (the {\em empty ball size} or {\em local feature size} measured atcertain points on the manifold), and have applications in Delaunaytriangulation-based algorithms for provably good surface reconstruction andprovably good mesh generation. Our bounds have better constants than the priorbounds we know of---and for several results in higher dimensions, our boundsare the first to give explicit constants.

Approximation Bounds for Interpolation and Normals on Triangulated Surfaces and Manifolds