The surface of the swallowtail represents the collection of points in parametric space corresponding to polynomials with double roots. Path Three is the curve of self-intersection. Therefore, the points on the curve belong to two parts of the swallowtail and correspond to polynomials with two double roots. As the point in parametric space approaches the origin, the double roots of the polynomial converge to a single quadruple root.
To see another path in parametric space which includes the origin, see Path Four.