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The cyclic quotient surface singularity $\frac{1}{r}(q,1)$ is the only singular point of a complete four-ray toric surface exactly under the displayed condition. Normalize the three smooth cones to the consecutive rays $(a,-1),(1,0),(0,1),(-1,b)$. The singular cone has determinant $1-ab=r$, so $ab=1-r$. In the basis $(-1,b),(0,-1)$, its other generator is $-a(-1,b)+r(0,-1)$, giving quotient parameter $a\mathrm{mod}r$. Thus $a=\pm s$ for a positive divisor $s$ of $r-1$. Conversely these rays construct the surface. Exchanging the two axes replaces $q$ by $q^{-1}$; replacing a factor by its complementary factor preserves the criterion under this equivalence.

A toric surface is a two-dimensional normal toric variety, described by a toric fan in a rank-two toric lattice. Primitive consecutive rays bound its affine charts; those charts are smooth exactly when the two ray generators form a lattice basis.