"When can you say that the curve is not linear anymore?"
That depends entirely on the problem, your requirements, any knowledge of the underlying system, the tools that you use, their numerical precision, the output that you need, the level of accuracy that you require, the further processing that might occur, the arbitrary demands of your customers and a million other possible factors.
To visualize the curve transition, subset the data for all/many contiguous data subsets, fit the required curve type to each subset, and then plot the least squares residuals (this might be multidimensional). If there is a clear knee to this curve, then you can pick the subset range that gives the widest range of data with a small enough residual to satisfy your demands. If there is no knee, then your hypothesis is false.
