Abstract: We address three related problems. The first problem is to change the volume of a solid by a prescribed amount, while minimizing Hausdorff error. This is important for compensating volume change due to smoothing, subdivision, or advection. The second problem is to preserve the individual areas of infinitely small chunks of a planar shape, as the shape is deformed to follow the gentle bending of a smooth spine (backbone) curve. This is important for bending or animating textured regions. The third problem is to generate consecutive offsets, where each unit element of the boundary sweeps the same region. This is important for constant material-removal rate during numerically controlled (NC) machining. For all three problems, we advocate a solution based on normal offsetting, where the offset distance is a function of local or global curvature measures. We discuss accuracy and smoothness of these solutions for models represented by triangle or quad meshes or, in 2D, by spine-aligned planar quads. We also explore the combination of local distance offsetting with a new selective smoothing process that reduces discontinuities and preserves curvature sign.