In spherical geometry
, a spherical lune (or biangle) is an area on a sphere
bounded by two half great circles
which meet at antipodal points
. It is an example of a digon
, with dihedral angle
The word "lune" derives from ''luna
'', the Latin
word for Moon.
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 circles are lines of longitude
(''meridians'') on a sphere, which meet at the north
and south pole
A spherical lune has two planes of symmetry. It can be bisected into two lunes of half the angle, or it can be bisected by an equatorial line into two right spherical triangles.
The surface area
of a spherical lune is 2θ ''R''2
, where ''R'' is the radius of the sphere and θ is the dihedral angle
in radians between the two half great circles.
When this angle equals 2π radians (360°) — i.e., when the second half great circle has moved a full circle, and the lune in between covers the sphere as a spherical monogon
— the area formula for the spherical lune gives 4π''R''2
, the surface area of the sphere
is a tessellation
of the sphere by lunes. A n-gonal regular hosohedron, has ''n'' equal lunes of π/''n'' radians. An ''n''-hosohedron has dihedral symmetry
(*22''n'') of order 4''n''. Each lune individually has cyclic symmetry
(*22) of order 4.
Each hosohedra can be divided by an equator
ial bisector into two equal spherical triangle
The visibly 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 unseen half. The spherical lune is a lighted crescent
shape seen from Earth.
_of_the_[[3-sphere's_parallels_(red),_[[Meridian_(perimetry,_visual_field).html" style="text-decoration: none;"class="mw-redirect" title="3-sphere.html" style="text-decoration: none;"class="mw-redirect" title="Stereographic projection of the [[3-sphere">Stereographic projection of the [[3-sphere's parallels (red), [[Meridian (perimetry, visual field)">meridians (blue) and hypermeridians (green). Lunes exist between pairs of blue meridian arcs.
Lunes can be defined on higher dimensional spheres as well.
In 4-dimensions a [[3-sphere is a generalized sphere. It can contain regular [[digon lunes as θ,φ
, where θ and φ are two dihedral angles.
For example, a regular hosotope
has digon faces, 2π/p,2π/q
, where its vertex figure
is a spherical platonic solid
, . Each vertex of defines an edge in the hosotope and adjacent pairs of those edges define lune faces. Or more specifically, the regular hosotope , has 2 vertices, 8 180° arc edges in a cube
, , vertex figure
between the two vertices, 12 lune faces, π/4,π/3
, between pairs of adjacent edges, and 6 hosohedral cells, π/3
* Beyer, W. H. ''CRC Standard Mathematical Tables'', 28th ed. Boca Raton, Florida: CRC Press, p. 130, 1987.
* Harris, J. W. and Stocker, H. "Spherical Wedge." §4.8.6 in ''Handbook of Mathematics and Computational Science.'' New York: Springer-Verlag, p. 108, 1998.
* Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; and Künstner, H. (Eds.). ''VNR Concise Encyclopedia of Mathematics'', 2nd ed. New York: Van Nostrand Reinhold, p. 262, 1989.