An example of a projective plane, that of order $2$, known as the Fano plane (for projective planes, order $q$ means $q+1$ points on each line, $q+1$ lines through each point):

An edge here is represented by a straight line, and the inscribed circle is also an edge. In other words, for a vertex set $\{1,2,3,4,5,6,7\}$, the edges of the Fano plane are

Notice that the Fano plane is generated by the triple $\{1,2,4\}$ by repeatedly adding $1$ to each entry, modulo $7$. The generating triple has the property that the differences of any two elements, in either order, are all pairwise different modulo $7$. In general, if we can find a set of $q+1$ of the integers (mod $q^{2}+q+1$) with all pairwise differences distinct, then this gives a cyclic representation of the finite plane of order $q$.