|In plane geometry, the crescent shape formed from two intersecting circles is called a lune (in gray). There are two lunes in each diagram only one of which is shaded|
In spherical geometry, a lune is an area on a sphere bounded by two half great circles, which is also called a digon or a diangle or (in German) a Zweieck. Great circles are the largest possible circles (circumferences) of a sphere; each one divides the surface of the sphere into two equal halves. Two great circles always intersect at two polar opposite points. Common examples of great quasi-circles are lines of longitude (meridians), which meet at the North and South Poles. Thus, the area between two meridians of longitude is a quasi-lune (slightly wider in the middle than a true lune due to the Earth being larger round the equator). The area of a spherical lune is 2θ R2, where R is the radius of the sphere and θ is the dihedral angle between the two half great circles. When this angle equals 2π — i.e., when the second half great circle has moved a full circle, and the lune in between covers the sphere — the area formula for the spherical lune gives 4πR2, the surface area of the sphere.
The lighted portion of the Moon visible from the Earth is a spherical lune. The first of the two intersecting great circles is the terminator between the sunlit half of the Moon and the dark half. The second great circle is a terrestrial terminator that separates the half visible from the Earth from the invisible half. This lighted, variable spherical (3D) lune is a crescent shape seen from Earth, the intersection of a semicircle and semi-ellipse (with the major axis of the ellipse coinciding with a diameter of the circle), as illustrated in the figure on the left.
Lune of Hippocrates
|Wikimedia Commons has media related to Crescents.|
- The Five Squarable Lunes at MathPages