Integer forward differencing of cubic polynomials

Two incremental cubic interpolation algorithms are derived and analysed. Each is based on a known linear interpolation algorithm and modified for third order forward differencing. The tradeoff between overflow avoidance and loss of precision has made forward differencing a method which, although known to be fast, can be difficult to implement. It is shown that there is one particular family of curves which represents the worst case, in the sense that if a member of this family can be accurately drawn without overflow, then any curve which fits in the bounding box of that curve can be. From this the limitations in terms of step count and screen resolution are found for each of the two algorithms.