In this paper, we present novel techniques to modify free-form shapes defined by implicit algebraic models. These models are first simplified by introducing a decomposition in terms of conic-line primitives and then the desired local/global shape changes are produced by varying the parameters of these primitives. A decomposed model provides efficient control of each primitive using its canonical frame. To make the analysis simpler, we will restrict our attention to free-form curves modeled by a 4th degree (quartic) equation. However, all of the analysis can be directly extended to higher degree models. We also provide some 3D interpolation procedures based on the resulting primitives to illustrate the potential of our methods in 3D.