We can simply apply the sweep-method found in the lecture. For reference, this method took an arbitrary point-set \(\in \mathbb{R}^2\) and found the convex hull. The run-time was comprised of an \(O(n log n)\) sort and an \(O(n)\) sweep. In our case we have already sorted our points as a pre-processing step. As such, we simply sweep on \(H_1 \cup H_2\) in \(O(|H_1| + |H_2|\) and set \(H\) to be the output of this sweep.