We will now slice the swallow tail and its tangent plane along the line v = 0 in order to show a cross-section thus verifying the tangency. (The black plane represents the cutting plane.)

In the above graphic, you can see that the orange plane is indeed tangent to the swallowtail along the line v = 0.

Having found a plane tangent to the swallowtail, let us now investigate the relationship between polynomials corresponding to points on the plane.

In the above graphic, the red point belongs to both the tangent plane and the swallowtail. The corresponding polynomial has a double root at x = 0.

In this graphic, the red point is on the tangent plane, but not on the swallowtail. The corresponding polynomial does not have a double root, but does have a single root at x = 0.

Similarly, the point in this graphic is not on the swallowtail, but is on the tangent plane. So, the corresponding polynomial has a root at x = 0.

This example illustrates that points on a tangent plane to the swallowtail share a common root. For a discussion of a similar phenomenon occuring with tangent lines to a bifurcation curve, see Project One.

Introduction
Overview
Path One
Path Two
Path Three
Path Four