Each cubic, monic polynomial of the form p(x) = x³ + a x + b is represented by a point (a, b) in parametric space. A polynomial will have a double root r if and only if p(r) = 0 and p'(r) = 0. The set of points (a, b) which correspond to polynomials with a double root form a boundary in parametric space between the points representing the cubics with three roots and those with one root. This boundary is a curve known as the bifurcation curve, B(a, b), such that a = -3 x² and b = 2 x³. The left side of the above graphic shows parametric space with B(a, b). The circle in parametric space represents an arbitrary collection of changing points (a, b) such that a² + b² = 0.25. As the parameters change, the corresponding polynomial in real space is shown changing on the right.

For each (a, b) point on the circle (and for all (a, b) points, of which the circle is an arbitrary collection), each root of the corresponding polynomial is shared by a certain polynomial whose (a_B, b_B) lies on the bifurcation curve. This is because the parametrization of the bifurcation curve was derived by setting p(x) = 0 and p'(x) = 0. Therefore, given any point on the circle (a₁, b₁) and the root x₁ of the corresponding polynomial, there will be one point (a_B, b_B) on the bifurcation curve whose polynomial will have the double root x₁. The line connecting (a₁, b₁) and (a_B, b_B) must contain only one point on the bifurcation curve. This is only possible if the line is tangent to the bifurcation curve. Refer to the above graphic for a representation of the tangent lines from each point on the circle to the points of a common root on the bifurcation curve. Click on the graphic to see the Mathematica program, which we wrote to create the graphic and which contains equations implied by the above explanation.