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As a core component of dispersed-dot halftoning, this paper focuses on the definition of new measures for giving or measuring good point distributions in a plane. By defining good point distributions from a purely geometric viewpoint of circle packing, it is shown that the energy defined by a certain strong convex function satisfies the necessary conditions for obtaining good point distributions in any point density by minimizing the energy. The energy with such the conditions are mathematically plain and there are no obscure parameters. The theory is also significantly motivated by a requirement of the adjustability to discrete spaces, and it is shown that the conditions actually work well also in the spaces. As an application, by using technically simple methods, dispersed-dot halftone masks are designed and goodness of point distributions of masks are estimated.