Spherical geometry

# Spherical geometry

Overview
Spherical geometry is the geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

of the two-dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

al surface of a sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

. It is an example of a geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

which is not Euclidean. Two practical applications of the principles of spherical geometry are to navigation
Navigation is the process of monitoring and controlling the movement of a craft or vehicle from one place to another. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks...

and astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

.

In plane geometry
Plane geometry
In mathematics, plane geometry may refer to:*Euclidean plane geometry, the geometry of plane figures,*geometry of a plane,or sometimes:*geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others;*geometry of the hyperbolic...

the basic concepts are point
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

s and (straight) line
Line (mathematics)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...

s. On the sphere, points are defined in the usual sense. The equivalents of lines are not defined in the usual sense of "straight line" in Euclidean geometry but in the sense of "the shortest paths between points," which are called geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

s.
Discussion

Encyclopedia
Spherical geometry is the geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

of the two-dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

al surface of a sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

. It is an example of a geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

which is not Euclidean. Two practical applications of the principles of spherical geometry are to navigation
Navigation is the process of monitoring and controlling the movement of a craft or vehicle from one place to another. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks...

and astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

.

In plane geometry
Plane geometry
In mathematics, plane geometry may refer to:*Euclidean plane geometry, the geometry of plane figures,*geometry of a plane,or sometimes:*geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others;*geometry of the hyperbolic...

the basic concepts are point
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

s and (straight) line
Line (mathematics)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...

s. On the sphere, points are defined in the usual sense. The equivalents of lines are not defined in the usual sense of "straight line" in Euclidean geometry but in the sense of "the shortest paths between points," which are called geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

s. On the sphere the geodesics are the great circle
Great circle
A great circle, also known as a Riemannian circle, of a sphere is the intersection of the sphere and a plane which passes through the center point of the sphere, as opposed to a general circle of a sphere where the plane is not required to pass through the center...

s; other geometric concepts are defined as in plane geometry but with straight lines replaced by great circles. Thus, in spherical geometry angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

s are defined between great circles, resulting in a spherical trigonometry
Spherical trigonometry
Spherical trigonometry is a branch of spherical geometry which deals with polygons on the sphere and the relationships between the sides and the angles...

that differs from ordinary trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

in many respects; for example, the sum of the interior angles of a triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

exceeds 180 degrees.

Spherical geometry is not elliptic geometry
Elliptic geometry
Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one...

but shares with that geometry the property that a line has no parallels through a given point. Contrast this with Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, in which a line has one parallel through a given point, and hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

, in which a line has two parallels and an infinite number of ultraparallels through a given point.

An important geometry related to that of the sphere is that of the real projective plane
Real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...

; it is obtained by identifying antipodal point
Antipodal point
In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite to it — so situated that a line drawn from the one to the other passes through the centre of the sphere and forms a true diameter....

s (pairs of opposite points) on the sphere. (This is elliptic geometry.) Locally, the projective plane has all the properties of spherical geometry, but it has different global properties. In particular, it is non-orientable
Orientability
In mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the...

, or one-sided.

Concepts of spherical geometry may also be applied to the oblong sphere, though minor modifications must be implemented on certain formulas.

Higher-dimensional spherical geometries exist; see elliptic geometry
Elliptic geometry
Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one...

.

## History

Spherical trigonometry was studied by early Greek mathematicians
Greek mathematics
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...

such as Menelaus of Alexandria
Menelaus of Alexandria
Menelaus of Alexandria was a Greek mathematician and astronomer, the first to recognize geodesics on a curved surface as natural analogs of straight lines.-Life and Works:...

, who wrote a book on spherical trigonometry called Sphaerica and developed Menelaus' theorem
Menelaus' theorem
Menelaus' theorem, named for Menelaus of Alexandria, is a theorem about triangles in plane geometry. Given a triangle ABC, and a transversal line that crosses BC, AC and AB at points D, E and F respectively, with D, E, and F distinct from A, B and C, thenThis equation uses signed lengths of...

.

### Islamic world

Muslims, according to Carra de Vaux, were "unquestionably the inventors of plane and spherical geometry, which did not, strictly speaking, exist among the Greeks". The book of unknown arcs of a sphere written by Islamic mathematician Al-Jayyani
Al-Jayyani
Abū ʿAbd Allāh Muḥammad ibn Muʿādh al-Jayyānī was a mathematician, Islamic scholar, and Qadi from Al-Andalus...

is considered to be the first treatise on spherical trigonometry. The book contains formulae for right-handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar triangle.

The book On Triangles by Regiomontanus
Regiomontanus
Johannes Müller von Königsberg , today best known by his Latin toponym Regiomontanus, was a German mathematician, astronomer, astrologer, translator and instrument maker....

, written around 1463, is the first pure trigonometrical work in Europe. However, Gerolamo Cardano
Gerolamo Cardano
Gerolamo Cardano was an Italian Renaissance mathematician, physician, astrologer and gambler...

noted a century later that much of the material there on spherical trigonometry was taken from the twelfth-century work of the Spanish Islamic scholar Jabir ibn Aflah
Jabir ibn Aflah
Abū Muḥammad Jābir ibn Aflaḥ was a Muslim astronomer and mathematician from Seville, who was active in 12th century Andalusia. His work Iṣlāḥ al-Majisṭi influenced Islamic, Jewish and Christian astronomers....

.

• SIGI
Sigi
In the Völsung cycle, Sigi is the ancestor of the Völsung lineage. In the Völsunga saga , he is said to be one of the sons of Odin. He is also listed among Odin's sons in the Nafnaþulur. He had a son called Rerir....

• Spherical distance
Great-circle distance
The great-circle distance or orthodromic distance is the shortest distance between any two points on the surface of a sphere measured along a path on the surface of the sphere . Because spherical geometry is rather different from ordinary Euclidean geometry, the equations for distance take on a...

• Spherical polyhedron
Spherical polyhedron
In mathematics, a spherical polyhedron is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons...

• Half-side formula
Half-side formula
In spherical trigonometry, the half side formula relates the angles and lengths of the sides of spherical triangles, which are triangles drawn on the surface of a sphere and so have curved sides and do not obey the formulas for plane triangles....