Self-intersection of nod(oid)al curves

We classify the self-intersections of a family of curves sharing the abstract properties of the meridian of a nodoid. For the purposes of this paper such a curve is termed a nodal curve, (in preference to the cumbersome nodoidal curve). The same properties are shared by certain Euler elastica and many other periodic curves. A careful definition is given defining the class of curves under consideration along with related examples. For nodal curves an order of self-intersection is defined. It is demonstrated that nodal curves can only have transverse double points of intersection lying within a family of evenly spaced parallel lines. Additional structure is also present which, for example, partitions all nodal curves into four distinct self-intersection types determined by the order of self-intersection. An explicit formula is given for the order. The location of an intersection point within a particular line of the parallel family also identifies two specific “loops” of the nodal curve determining the intersection.