Optimized Hemisphere Point Sets with Antipodal Completion: Angular Covering Radius

Dataset overview This dataset provides 1,196 numerically optimized discrete point sets on the closed upper unit hemisphere, with one point set for every integer N from 5 to 1,200. Each stored point is a three-dimensional Cartesian unit vector with a non-negative third coordinate. Each stored hemisphere point set is evaluated through its antipodal completion, thereby providing nearly uniform directional coverage of the entire unit sphere. Optimization objective For each value of N, the point configuration was numerically optimized to reduce its angular covering radius. The angular covering radius is the largest angular distance between an arbitrary point on the unit sphere and its nearest representative in the antipodally completed point set. It therefore quantifies the worst-case angular discretization error. A smaller covering radius indicates better coverage of the unit sphere. Potential applications discretization of orientation-dependent objective functions; representation and selection of unoriented plane normals; directional search and optimization on the hemisphere; reduction of worst-case angular mismatch; benchmarking of spherical and projective point-distribution methods. Data generation Initial point configurations were generated using a spherical Fibonacci distribution based on the golden angle. The configurations were then refined using a Riesz-energy objective, maximization of the convex-hull volume, and final optimization of the angular covering radius. The optimization was performed in MATLAB using fminunc in the Optimization Toolbox. Candidate point configurations were compared across successive values of N so that the retained angular covering radius does not increase as the number of stored hemisphere points increases. Included material MATLAB MAT-files containing the point coordinates, point count, and angular covering radius; headerless CSV files containing Cartesian coordinates x, y, and z; importable Python/NumPy modules; C source files containing separate constant coordinate arrays; a summary table containing the covering radius and the improvement between consecutive point-set sizes; format-specific README files; a detailed dataset report describing the mathematical formulation, data-generation procedure, file structure, validation, and limitations; selected visualizations of the point distributions and their antipodal completions. Quality and limitations The point sets are optimized for angular covering. The associated angular covering radius is reported for every point-set size. Although the configurations were obtained through extensive numerical optimization, they have not been certified as globally optimal. Consequently, configurations with smaller angular covering radii may exist. AI assistance disclosure Generative AI was used to assist with language editing and the organization of the accompanying documentation. The generation of the point sets, the numerical results, and responsibility for the scientific content remain with the authors.