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In $PG(2,q^3)$, let $\pi$ be a subplane of order $q$ that is tangent to$\ell_infty$. The tangent splash of $\pi$ is defined to be the set of $q^2+1$points on $\ell_infty$ that lie on a line of $\pi$. This article investigatesproperties of the tangent splash. We show that all tangent splashes areprojectively equivalent, investigate sublines contained in a tangent splash,and consider the structure of a tangent splash in the Bruck-Bose representationof $PG(2,q^3)$ in $PG(6,q)$. We show that a tangent splash of $PG(1,q^3)$ is a$GF(q)$-linear set of rank 3 and size $q^2+1$; this allows us to use resultsabout linear sets from \cite{lavr10} to obtain properties of tangent splashes.

An investigation of the tangent splash of a subplane of PG(2,q^3)