Suppose **f crosses C when **f. We now have certain squares with three corners on C. For any such square the middle corner of these will be called the vertex of the square and the corner not on the curve will be called the diagonal point of the square. Each point on C, as a vertex, may possess a finite number of corresponding diagonal points by the above construction.

To each paired vertex and diagonal point there corresponds a unique forward corner point, i.e., the corner on C reached first by proceeding along C from the vertex in the direction of increasing t. If the vertex is at **f, and if the interior of C is on the left as one moves in the direction of increasing t, then every such corner can be found from the curve obtained by rotating C clockwise through 90 ` about the vertex. The set of intersections of **f, the rotated curve, with the original curve C consists of just the set of forward corner points on C corresponding to the vertex at **f, plus the vertex itself. We note that two such curves C and **f, cannot coincide at more than a finite number of points; otherwise, being analytic, they would coincide at all points, which is impossible since they do not coincide near **f.

With each vertex we associate certain numerical values, namely the set of positive differences in the parameter t between the vertex and its corresponding forward corner points. For the vertex at **f, these values will be denoted by **f. The function f (t) defined in this way is multi valued.