PLANE CURVE DIAGRAMS AND GEOMETRICAL APPLICATIONS

We look at plane curve diagrams (f,α), which are given by a plane curve multigerm α : (ℝ, S) → ℝ² and a function on it f:(ℝ, S) → ℝ. We obtain a classification of all such diagrams, where α hase-codimension ≤ 2 and f has finite order. Then we define an equivalence between plane curves which we callh-equivalence and which is determined by the class of the diagram (hα, α). Here, hα denotes the height function of α with respect to its normal vector. This is an equivalence which not only takes into account the topology of the singularity of α, but also its flat geometry. Finally, we apply our results in order to obtain a classification of all the plane projections of a generic space curve γ embedded in ℝ³