Abstract: Several design and animation techniques use a one-dimensional proxy C (a spine curve in 3D) to control the deformation or behavior of a digital model of a 3D shape S. We propose a modification of these "skinning" techniques that ensures local volume preservation, which is important for the physical plausibility of digital simulations. In the proposed "fleshing" techniques, as input, we consider a smooth spine C0, a model S0 of a solid that lies "sufficiently close" to C0, and a deformed version C1 of C0 that is "not overly bent". (We provide a precise characterization of these restrictions.) As output, we produce a bijective mapping M, that maps any point X of S onto a point M(X) of M(S). M satisfies two properties: (1) The closest projection of X on C0 and of M(X) on C1 have the same arc length parameter. (2) U and M(U) have the same volume, where U is any subset of S. We provide three different closed form expressions for radial, normal and binormal fleshing and discuss the details of their practical real-time implementation.