The paths of the roots of the quadratic function II

Abstract

In Yamaoka (2023) we fixed two real coefficients and varied the remaining real coefficient of the quadratic function to obtain the paths described by the two roots of the function in the $\C$ plane. In this article, we used the inverse $\Phi$ of the stereographic projection to take the paths of both roots to the sphere $\mathbb{S}^2 \subset \R^3$: except in the subcase in which the paths of the images by $\Phi$ of the two roots are restricted to the south pole $S$ of $\mathbb{S}^2$, in each of the other subcases the paths of the images by $\Phi$ of both roots rest on one or two circumferences on $\mathbb{S}^2$. As the variable coefficient approaches $-\infty$ and $+\infty$, we observed the relations that the two roots have, via $\Phi$, with the north $N$ and/or south $S$ poles of $\mathbb{S}^2$. Let $G_j \subset \mathbb{S}^2$ be the set of points of the path of the image by $\Phi$ of the root $j$ on $\mathbb{S}^2$, $j=1,2$. We determined the common adherent points to $G_1$ and to $G_2$. We used Differential Calculus, Analytical Geometry with vector treatment, the inverse of stereographic projection and the distance between two sets.