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 on a sphere; each great circle divides the surface of the sphere into two equal halves. Two great circles always intersect at two polar opposite points. Common examples of great circles are lines of longitude (meridians), which meet at the North and South Poles. Thus, the area between two meridians of longitude is a lune. 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 that which separates the half visible from the Earth from the invisible half. This lighted spherical lune produces the familiar crescent shape of the Moon seen from Earth, the intersection of a semicircle and semi-ellipse (with the major axis of the ellipse coinciding with the diameter of the circle), as illustrated in the figure on the left.
Lune of Hippocrates
- The Five Squarable Lunes at MathPages
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