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In this paper we propose a new discrete differential error metric for surface simplification. Many surface simplification algorithms have been developed in order to produce rapidly high quality approximations of polygonal models, and the quadric error metric based on the distance error is the most popular and successful error metric so far Even though such distance based error metrics give visually pleasing results with a reasonably fast speed, it is hard to measure an accurate geometric error on a highly curved and thin region since the error measured by the distance metric on such a region is usually small and causes a loss of visually important features. To overcome such a drawback, we define a new error metric based on the theory of local differential geometry in such a way that the first and the second order discrete differentials approximated locally on a discrete polygonal surface are integrated into the usual distance error metric. The benefits of our error metric are preservation of sharp feature regions after a drastic simplification, small geometric errors, and fast computation comparable to the existing methods.