In this thesis, we present a new approach to rendering deforming textured surfaces that takes into account variations in elasticity of the materials represented in the texture. Our approach is based on dynamically warping the parameterisation so that parameterisation distortion in a deformed pose is locally similar to the rest pose; this similarity results in apparent rigidity of the mapped texture material. The warps are also weighted, so that users have control over what appears rigid and what not. Our algorithms achieve real-time generation of warps, including their application in rendering the textured surfaces. A key factor to the achieved performance is the exploitation of the parallel nature of local optimisations by implementing the algorithms on the GPU. We demonstrate our approach with several example applications. We show warps on models using standard texture mapping as well as Ptex. We also show warps using static or dynamic/procedural texture detail, while the surface that it is mapped on deforms. A variety of use-cases is also provided: generating warps for looping animations, generating out-of-core warps of film-quality assets, approximating high-resolution warps with lower-resolution texture-space Linear Blend Skinning and dynamically preserving texture features of a model being interactively edited by an artist.