We study the position distribution of a single active Brownian particle (ABP) on the plane. We show that this distribution has a compact support, the boundary of which is an expanding circle. We focus on a short-time regime and employ the optimal fluctuation method (OFM) to study large deviations of the particle position coordinates $x$ and $y$. We determine the optimal paths of the ABP, conditioned on reaching specified values of $x$ and $y$, and the large deviation functions of the marginal distributions of $x$, and of $y$. These marginal distributions match continuously with near tails of the $x$ and $y$ distributions of typical fluctuations, studied earlier. We also calculate the large deviation function of the joint $x$ and $y$ distribution $P(x,y,t)$ in a vicinity of a special zero-noise point, and show that $ln P(x,y,t)$ has a nontrivial self-similar structure as a function of $x$, $y$ and $t$. The joint distribution vanishes extremely fast at the expanding circle, exhibiting an essential singularity there. This singularity is inherited by the marginal $x$- and $y$-distributions. We argue that this fingerprint of the short-time dynamics remains there at all times.
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