In an increasing number of applications triangle meshes represent a flexible and efficient alternative to traditional NURBS-based surface representations. Especially in engineering applications it is crucial to guarantee that a prescribed approximation tolerance to a given reference geometry is respected for any combination of geometric algorithms that are applied when processing a triangle mesh. We propose a simple and generic method for computing the distance of a given polygonal mesh to the reference surface, based on a linear approximation of its signed distance field. Exploiting the hardware acceleration of modern GPUs allows us to perform up to 3M triangle checks per second, enabling real-time distance evaluations even for complex geometries. An additional feature of our approach is the accurate high-quality distance visualization of dynamically changing meshes at a rate of 15M triangles per second. Due to its generality, the presented approach can be used to enhance any mesh processing method by global error control, guaranteeing the resulting mesh to stay within a prescribed error tolerance. The application examples that we present include mesh decimation, mesh smoothing and freeform mesh deformation.