  Let $\mathbb{k}$ be an algebraically closed field. Building upon previous
work, we classify, up to isomorphism of graded algebras, quadratic graded
twisted tensor products of $\mathbb{k}[x,y]$ and $\mathbb{k}[z]$. When such an
algebra is Artin-Schelter regular, we identify its point scheme and type. We
also describe which three-dimensional Sklyanin algebras contain a subalgebra
isomorphic to a quantum $\mathbb{P}^1$, and we show that every algebra in this
family is a graded twisted tensor product of $\mathbb{k}_{-1}[x,y]$ and
$\mathbb{k}[z]$.
