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

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