  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.
