Spherical Lune on:
[Wikipedia]
[Google]
[Amazon]
In
The
The visibly lighted portion of the
spherical geometry
300px, A sphere with a spherical triangle on it.
Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sp ...
, a spherical lune (or biangle) is an area on a sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
bounded by two half great circles which meet at antipodal points
In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true ...
. It is an example of a digon
In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visu ...
, θ, with dihedral angle
A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the un ...
θ. The word "lune" derives from ''luna
Luna commonly refers to:
* Earth's Moon, named "Luna" in Latin
* Luna (goddess), the ancient Roman personification of the Moon
Luna may also refer to:
Places Philippines
* Luna, Apayao
* Luna, Isabela
* Luna, La Union
* Luna, San Jose
Roma ...
'', the Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
word for Moon.
Properties
Great circles are the largest possible circles (circumferences) of asphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
; 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 let ...
(''meridians'') on a sphere, which meet at the north
North is one of the four compass points or cardinal directions. It is the opposite of south and is perpendicular to east and west. ''North'' is a noun, adjective, or adverb indicating direction or geography.
Etymology
The word ''north ...
and south pole
The South Pole, also known as the Geographic South Pole, Terrestrial South Pole or 90th Parallel South, is one of the two points where Earth's axis of rotation intersects its surface. It is the southernmost point on Earth and lies antipod ...
s.
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.
Surface area
The surface area
The surface area 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 definition of ...
of a spherical lune is 2θ ''R''2, where ''R'' is the radius of the sphere and θ is the dihedral angle
A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the un ...
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.
Examples
Ahosohedron
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 ...
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
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binar ...
''C2v'', (*22) of order 4.
Each hosohedra can be divided by an equator
The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can also ...
ial bisector into two equal 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 ...
s.
Astronomy
The visibly lighted portion of the Moon
The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
visible from the Earth is a spherical lune. The first of the two intersecting great circles is the terminator
Terminator may refer to:
Science and technology
Genetics
* Terminator (genetics), the end of a gene for transcription
* Terminator technology, proposed methods for restricting the use of genetically modified plants by causing second generation s ...
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 in the first quarter (the "sickle moon"), or by extension a symbol representing the Moon itself.
In Hinduism, Lord Shiva is often shown wearing a crescent moon on his ...
shape seen from Earth.
''n''-sphere lunes
Lunes can be defined on higher dimensional spheres as well. In 4-dimensions a3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensio ...
is a generalized sphere. It can contain regular digon
In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visu ...
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 polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
is a spherical platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
, . 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
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only ...
, , vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
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