Focus+Context techniques are commonly used in visualization systems to simultaneously provide both the details and the context of a particular dataset. This paper proposes a new methodology to empirically investigate the effect of various Focus+Context transformations on human perception. This methodology is based on the shaker paradigm, which tests performance for a visual task on an image that is rapidly alternated with a transformed version of itself. An important aspect of this technique is that it can determine two different kinds of perceptual cost: (i) the effect on the perception of a static transformed image, and (ii) the effect of the dynamics of the transformation itself. This technique has been successfully applied to determine the extent to which human perception is invariant to scaling and rotation [Rensink 2004]. In this paper, we extend this approach to examine nonlinear fisheye transformations of the type typically used in a Focus+Context system. We show that there exists a no-cost zone where performance is unaffected by an abrupt, noticeable fisheye transformation, and that its extent can be determined. The lack of perceptual cost in regards to these sudden changes contradicts the belief that they are necessarily detrimental to performance, and suggests that smoothly animated transformations between visual states are not always necessary. We show that this technique also can map out low-cost zones where transformations result in only a slight degradation of performance. Finally, we show that rectangular grids have no positive effect on performance, acting only as a form of visual clutter. These results therefore demonstrate that the perceptual costs of nonlinear transformations can be successfully quantified. Interestingly, they show that some kinds of sudden transformation can be experienced with minimal or no perceptual cost. This contradicts the belief that sudden changes are necessarily detrimental to performance, and suggests that smoothly animated transformations between visual states are not always necessary...