This thesis analyzes an optimal algorithm for the half-plane hitting set problem in the plane. Given a set P of points and a set H of half-planes, the goal is to compute a smallest subset P ′ ⊆ P such that every half-plane in H contains at least one point of P ′. This problem is a fundamental geometric optimization task with links to coverage, duality, and algorithm design. Earlier work solved the general problem in O(n³ log n) time, while a lower bound of Ω(n log n) was known [4]. The algorithm studied in this thesis closes this gap by achieving O(n log n) time, which is asymptotically optimal. The key contribution is a reduction of the geometric hitting-set instance to a circular-point coverage problem, followed by a structural improvement that replaces a quadratic-size arc set with a linear-size representative subset Â; the sufficiency of this reduction is formalized in Lemma 2. The thesis first introduces the formal model, assumptions, and lower-bound context, then explains prior approaches and why they are not optimal for the general setting. It then presents the reduction, proves correctness via central lemmas and a corollary, and derives both intermediate and final time bounds. Special attention is given to design choices, bottlenecks, and the role of envelope-based pruning and data-structural acceleration in obtaining linear-size candidate families. The result shows that the half-plane hitting set problem can be solved as fast as comparison-based sorting in the algebraic decision-tree model. Beyond proving optimal asymptotic complexity, the analyzed approach is notable for conceptual clarity compared with earlier multi-instance reductions. The thesis concludes with related problems (coverage and weighted variants), limitations, and possible directions for future research.