This paper introduces a unified and flexible set of tools to approximate geometric attributes such as normal vectors and curvatures on arbitrary triangle meshes. The authors present a consistent derivation of these first and second-order differential properties using averaging Voronoi cells and the mixed Finite-Element/Finite-Volume method, comparing them to existing formulations. The operators are closely related to their continuous counterparts, ensuring appropriate extension from continuous to discrete settings. They demonstrate that these estimates are optimal under mild smoothness conditions and showcase their numerical quality. The paper also presents applications of these operators, including mesh smoothing, enhancement, and quality checking, and shows results of denoising in higher dimensions, such as for tensor images. The contributions include defining and deriving first and second-order differential attributes for piecewise linear surfaces, presenting a unified framework for accurate and robust computation, and providing numerical results and applications.This paper introduces a unified and flexible set of tools to approximate geometric attributes such as normal vectors and curvatures on arbitrary triangle meshes. The authors present a consistent derivation of these first and second-order differential properties using averaging Voronoi cells and the mixed Finite-Element/Finite-Volume method, comparing them to existing formulations. The operators are closely related to their continuous counterparts, ensuring appropriate extension from continuous to discrete settings. They demonstrate that these estimates are optimal under mild smoothness conditions and showcase their numerical quality. The paper also presents applications of these operators, including mesh smoothing, enhancement, and quality checking, and shows results of denoising in higher dimensions, such as for tensor images. The contributions include defining and deriving first and second-order differential attributes for piecewise linear surfaces, presenting a unified framework for accurate and robust computation, and providing numerical results and applications.