Affine cubic functions

Abstract

The classification of affine cubic functions in the real case is a fairly easy corollary of that in the complex case (9). However as the results can be easily interpreted by diagrams, one can obtain a much richer understanding. For example, the question of which types of cubic curve occur as level curves of which types of function is now much less trivial. This will lead us first to re-examine the classification of cubic curves going back to Newton (4). Next the ‘dynamic’ approach of considering these curves as members of families leads to the diagrams associated with the umbilic catastrophes of Thorn (8). However the consideration of functions rather than of curves gives a 1-dimensional foliation of these diagrams which we describe next. We conclude by placing the results back in a protective setting.