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

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 $\mathbb{R}^\ell$ ($\ell \geq3$) 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.