Circular planar nearrings: geometrical and combinatorial aspects

Let $(N,\mathrm{\Phi })$ be a circular Ferrero pair. We define the disk with center $b$ and radius $a$, $\mathcal{D}(a;b)$, as $$\mathcal{D}(a;b)=\{x\in \mathrm{\Phi }(r)+c\mid r\mathrm{\ne }0,b\in \mathrm{\Phi }(r)+c,\mathrm{\mid }(\mathrm{\Phi }(r)+c)\cap (\mathrm{\Phi }(a)+b)\mathrm{\mid }=1\}\mathrm{.}$$ We prove that in the field-generated case there are many analogies with the Euclidean geometry. Moreover, if ${\mathcal{B}}^{\mathcal{D}}$ is the set of all disks, then, in some interesting cases, we show that the incidence structure $(N,{\mathcal{B}}^{\mathcal{D}},\in )$ is actually a balanced incomplete block design.