Geometric interpretation of generalized distance-squared mappings of ${\mathbb{R}}^2$ into ${\mathbb{R}}^{\ell }$ $(\ell \ge 3)$

Abstract

Generalized distance-squared mappings are quadratic mappings of ${\mathbb{R}}^m$ into ${\mathbb{R}}^{\ell }$ of special type. In the case that matrices $A$ constructed by coefficients of generalized distance-squared mappings of ${\mathbb{R}}^2$ into ${\mathbb{R}}^{\ell }$ ($\ell \ge 3$) are full rank, the generalized distance-squared mappings having a generic central point have the following properties. In the case of $\ell =3$, they have only one singular point. On the other hand, in the case of $\ell >3$, they have no singular points. Hence, in this paper, the reason why in the case of $\ell =3$ (resp., in the case of $\ell >3$), they have only one singular point (resp., no singular points) is explained by giving a geometric interpretation to these phenomena.

6 pages, 2 figures