Lune (mathematics)

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A lune is either of two figures, both shaped roughly like a crescent Moon. The word "lune" derives from luna, the Latin word for Moon.

1 Plane geometry
2 Spherical geometry
3 See also
4 References
5 External links

[edit] Plane geometry

In plane geometry, a lune is a concave area bounded by two arcs. The corresponding convex shape is a lens.

Formally, a lune is the relative complement of one circle in another (where they intersect but neither is a subset of the other).^[1] Alternatively, if A and B are circles, then $L = A - A \cap B$ is a lune.



In plane geometry, the crescent shape formed from two intersecting circles is called a lune (in gray).

[edit] Spherical geometry

A spherical lune. The two great circles are shown as thin black lines, whereas the lune itself (shown in green) is outlined in thick black lines, corresponding to its defining half great circles. The great circles intersect at two polar opposite points, such as the North and South poles.

In spherical geometry, a lune is an area on a sphere bounded by two half great circles,^[2] 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 crescent Moon is a spherical lune, as described in the text. Here, the blue and red portions may be taken equally as the lighted and dark portions of the Moon visible from Earth, or vice versa.

The lighted portion of the Moon is a spherical lune. The first of the two intersecting great circles is the boundary separating the lighted half of the Moon from the dark half. The second great circle is that which separates the half visible from the Earth from the invisible half. Seen face on, this lighted spherical lune produces the familiar crescent shape of the Moon seen from Earth, as illustrated in the Figure at the left.