The left side of the above graphic shows the swallowtail in parametric space. The circle in parametric space represents an arbitrary collection of changing points (a, b, c). As the point traverses the different regions of parametric space, the polynomial is shown changing on the right.
Notice that when the point is above the swallowtail, the polynomial has no real roots. When the point is below the swallowtail, the polynomial has two real roots. When the point is inside the space defined by the swallowtail's self-intersection, the polynomial has four real roots. In going between these regions, the point crosses the swallowtail (which is the collection of points corresponding to polynomials with double roots). At the self-intersection, the point represents a polynomial with two double roots; on the surface beneath the self-intersection, one double root and two single roots; and on the surface of the wings, one double root.
When the point crosses the surface of the wings, the double root of the polynomial appears to be a quadruple root. To convince yourself of its classification as a double root, see Path Two.