spherical geometry 300px, A sphere with a spherical triangle on it. Spherical geometry or spherics () is the geometry of the two-dimensional surface of a sphere or the -dimensional surface of higher dimensional spheres. Long studied for its practical applicati ...

, a spherical lune (or biangle) is an area on a

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

bounded by two half

great circles In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Discussion Any arc of a great circle is a geodesic of the sphere, so that great circles in spheric ...

which meet at

antipodal points In mathematics, two points of a sphere (or n-sphere, including a circle) are called antipodal or diametrically opposite if they are the endpoints of a diameter, a straight line segment between two points on a sphere and passing through its cent ...

. It is an example of a

digon In geometry, a bigon, digon, or a ''2''-gon, is a polygon with two sides (edge (geometry), edges) and two Vertex (geometry), vertices. Its construction is Degeneracy (mathematics), degenerate in a Euclidean plane because either the two sides wou ...

, _θ, with dihedral angle θ. The word "lune" derives from ''

luna Luna commonly refers to: * Earth's Moon, named "Luna" in Latin, Spanish and other languages * Luna (goddess) In Sabine and ancient Roman religion and myth, Luna is the divine embodiment of the Moon (Latin ''Lūna'' ). She is often presented as t ...

'', the

Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...

word for Moon.

Properties

Great circles are the largest possible circles (circumferences) of a

; 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 Longitude (, ) is a geographic coordinate that specifies the east- west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek lett ...

(''meridians'') on a sphere, which meet at the north and south poles. 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 triangle Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gre ...

Surface area

The

surface area The surface area (symbol ''A'') of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the d ...

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 In geometry, a monogon, also known as a henagon, is a polygon with one Edge (geometry), edge and one Vertex (geometry), vertex. It has Schläfli symbol .Coxeter, ''Introduction to geometry'', 1969, Second edition, sec 21.3 ''Regular maps'', p. 386 ...

— the area formula for the spherical lune gives 4π''R''², the surface area of the sphere.

Examples

hosohedron In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular -gonal hosohedron has Schläfli symbol with each spherical lune ha ...

is a

tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...

of the sphere by lunes. A n-gonal regular hosohedron, has ''n'' equal lunes of π/''n'' radians. An ''n''-hosohedron has

dihedral symmetry In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, g ...

D_''n''h, 'n'',2 (*22''n'') of order 4''n''. Each lune individually has cyclic symmetry ''C_2v'', (*22) of order 4. Each hosohedra can be divided by an

equator The equator is the circle of latitude that divides Earth into the Northern Hemisphere, Northern and Southern Hemisphere, Southern Hemispheres of Earth, hemispheres. It is an imaginary line located at 0 degrees latitude, about in circumferen ...

ial bisector into two equal

Astronomy

The visibly lighted portion of the

Moon The Moon is Earth's only natural satellite. It Orbit of the Moon, orbits around Earth at Lunar distance, an average distance of (; about 30 times Earth diameter, Earth's diameter). The Moon rotation, rotates, with a rotation period (lunar ...

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 A crescent shape (, ) is a symbol or emblem used to represent the lunar phase (as it appears in the northern hemisphere) in the first quarter (the "sickle moon"), or by extension a symbol representing the Moon itself. In Hindu iconography, Hind ...

shape seen from Earth.

''n''-sphere lunes

Lunes can be defined on higher dimensional spheres as well. In 4-dimensions a

3-sphere In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere, ''n''-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point. The interior o ...

is a generalized sphere. It can contain regular

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 In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...

is a spherical

platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...

, . 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 A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

, ,

between the two vertices, 12 lune faces, _π/4,π/3, between pairs of adjacent edges, and 6 hosohedral cells, _π/3.

References

{{reflist * 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. Spherical geometry