Details

The Coxeter group acts on the plane by permuting the coordinates. The space of its complex orbits can be identified with the space of polynomials , which is a smooth manifold. The Vieta map , , …, sends the space where the reflection group acts to the space of its orbits. The critical set of the Vieta map is the union of its mirrors. The variety of irregular orbits of a reflection group (orbits with less than the maximal number of points) are called its discriminant. The discriminant of the group shown in this Demonstration is the semicubical parabola in the target plane of the Vieta map. The image of the real plane under the Vieta mapping is the interior of the semicubical parabola.

Reference

[1] V. I. Arnold, The Theory of Singularities and Its Applications, New York: Lezioni Fermiane, 1993.