Shape transformation between objects of different topology and positions in space is an open modelling problem. We propose a new approach to solving this problem for two given 2D or 3D shapes. The key steps of the proposed algorithm are: increase dimension by converting two input kD shapes into half-cylinders in (k + 1)D space-time, applying bounded blending with added material to the half-cylinders, and making cross-sections for getting intermediate shapes under the transformation. The additional dimension is considered as a time coordinate for making animation. We use the bounded blending set operations in space-time defined using R-functions and displacement functions with the localized area of influence applied to the functionally defined half-cylinders. The proposed approach is general enough to handle input shapes with arbitrary topology defined as polygonal objects with holes and disjoint components, set-theoretic objects, or analytically defined implicit surfaces. The obtained unusual amoeba-like behaviour of the shape combines metamorphosis with the non-linear motion.