Representation of the Crescent Moon

The moon is the only celestial body who’s waxing and waning can be seen with the naked eye, and the crescent moon is often used to symbolize the moon itself. If we approximate the moon as a sphere and the sun's rays as parallel rays, the outer contour of the orthogonal projection of the crescent moon will be a semicircle (arc angle = 180°), and the inner contour will be a concentric elliptical arc. However, in the expressions of the crescent moon that are widely used, the arc angle of the outer contour is often larger than 180°, and the inner contour is often represented by an eccentric arc. In this paper, we show some of these expressions and discuss the representation of the crescent moon from perspectives of geometry of projection, human eyesight, and visual design. The principal results of the discussion are as follows; (1) although the moon shines beyond the hemisphere, it is too small to be detected by human eyesight, and (2) although the expression of the crescent moon shining beyond a semicircle differs from the human visual image, it is used by emphasizing the characteristic feature of the crescent moon and giving a sense of completion and stability.