Ringed planet, with planet radius s and outer ring radius c viewed from the direction of the star, defining φ (left side) and from the polar north , defining the angle φ′ (right side). The illuminated area consists of A plus C. On the left, A is the difference between two half ellipses with areas A + B and B; and C the projected circle segment C + D minus a circle segment D (with respective areas φ′c²sinβ and φs²).