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Spherical geometry
 
 
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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 mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
al surface of a sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
. It is an example of a non-Euclidean geometry
Non-Euclidean geometry

In mathematics, non-Euclidean geometry describes hyperbolic geometry and elliptic geometry, which are contrasted with Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of Parallel lines....
. Two practical applications of the principles of spherical geometry are navigation
Navigation

Navigation is the process of reading, 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 the science of Astronomical object and Phenomenon that originate outside the Earth's atmosphere . It is concerned with the evolution, physics, chemistry, meteorology, and motion of celestial objects, as well as the physical cosmology....
.

In plane geometry
Plane geometry

In mathematics, plane geometry may mean:*geometry of a plane ,*geometry of the Euclidean plane,or sometimes a plane is any flat surface that extends without end in all directions....
 the basic concepts are point
Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue....
s and line
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
. On the sphere, points are defined in the usual sense. The equivalents of lines are not defined in the usual sense of "straight line" but in the sense of "the shortest paths between points" which is called a geodesic
Geodesic

In mathematics, a geodesic [jee-uh-des-ik, -dee-sik] is a generalization of the notion of a "Line " to "manifolds".In presence of a Metric , geodesics are defined to be the shortest path between points on the space....
.






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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 mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
al surface of a sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
. It is an example of a non-Euclidean geometry
Non-Euclidean geometry

In mathematics, non-Euclidean geometry describes hyperbolic geometry and elliptic geometry, which are contrasted with Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of Parallel lines....
. Two practical applications of the principles of spherical geometry are navigation
Navigation

Navigation is the process of reading, 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 the science of Astronomical object and Phenomenon that originate outside the Earth's atmosphere . It is concerned with the evolution, physics, chemistry, meteorology, and motion of celestial objects, as well as the physical cosmology....
.

In plane geometry
Plane geometry

In mathematics, plane geometry may mean:*geometry of a plane ,*geometry of the Euclidean plane,or sometimes a plane is any flat surface that extends without end in all directions....
 the basic concepts are point
Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue....
s and line
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
. On the sphere, points are defined in the usual sense. The equivalents of lines are not defined in the usual sense of "straight line" but in the sense of "the shortest paths between points" which is called a geodesic
Geodesic

In mathematics, a geodesic [jee-uh-des-ik, -dee-sik] is a generalization of the notion of a "Line " to "manifolds".In presence of a Metric , geodesics are defined to be the shortest path between points on the space....
. On the sphere the geodesics are the great circle
Great circle

A great circle of a sphere is a circle that runs along the surface of that sphere so as to cut it into two equal halves. The great circle therefore has both the same circumference and the same center as the sphere....
s, so the other geometric concepts are defined like in plane geometry but with lines replaced by great circles. Thus, in spherical geometry angle
Angle

In geometry and trigonometry, an angle is the figure formed by two Ray sharing a common endpoint, called the vertex of the angle . The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide...
s are defined between great circles, resulting in a spherical trigonometry
Spherical trigonometry

Spherical trigonometry is a part of spherical geometry that deals with polygons on the sphere and explains how to find relations between the involved angles....
 that differs from ordinary trigonometry
Trigonometry

Trigonometry is a branch of mathematics that deals with triangle s, particularly those plane triangles in which one angle has 90 degrees . Trigonometry deals with relationships between the sides and the angles of triangles and with the trigonometric functions, which describe those relationships....
 in many respects (for example, the sum of the interior angles of a triangle exceeds 180 degrees).

Spherical geometry is the simplest model of 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....
, in which a line has no parallels through a given point. Contrast this with hyperbolic geometry
Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced. The parallel postulate in Euclidean geometry is equivalent to the statement that, in two dimensional space, for any given line l and point P not on l, there is exactly one line through P th...
, in which a line has two parallels, and an infinite number of ultra-parallels, through a given point.

An important related geometry related to that modeled by the sphere is called the real projective plane
Real projective plane

In mathematics, the real projective plane is the space of lines in R3 passing through the origin. It is a non-Orientability two-dimensional manifold, that is, a surface, that has basic applications to geometry, but which cannot be embedding in our usual three-dimensional space without intersecting itself....
; it is obtained by identifying antipodes
Antipodal point

In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diameter opposite 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....
 (pairs of opposite points) on the sphere. Locally, the projective plane has all the properties of spherical geometry, but it has different global properties. In particular, it is non-orientable
Orientability

A surface S in the Euclidean space R3 is orientable if a two-dimensional figure cannot be moved around the surface and back to where it started so that it looks like its own mirror image ....
. Concepts of spherical geometry may also be applied to the oblong sphere, though minor modifications must be implemented on certain formulas.

History

The book On Triangles by Regiomontanus is the first pure trigonometrical work in Europe, written around 1463, however Gerolamo Cardano
Gerolamo Cardano

Gerolamo Cardano or Girolamo Cardano was an Italy 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

Abu Muhammad Jabir ibn Aflah was an Arab Islamic astronomy, Islamic mathematics and Inventions in the Islamic world whose works, once translated into Latin , influenced later European mathematicians and astronomers....
.

The book of unknown arcs of a sphere

The book of unknown arcs of a sphere written by Islamic mathematician Al-Jayyani
Al-Jayyani

Abu Abd Allah Muhammad ibn Muadh Al-Jayyani, shortened to Al-Jayyani was an Arab Islamic mathematics from Al-Andalus . Al-Jayyani wrote important commentaries on Euclid's Euclid's Elements and he wrote the first treatise on spherical trigonometry in its modern form....
 is considered to be the first treatise on spherical trigonometry. His 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.

See also

  • SIGI
    Sigi

    In the V?lsung cycle, Sigi is the ancestor of the V?lsung lineage.In the V?lsunga saga , he is said to be the Sons of Odin....
  • Spherical distance
    Great-circle distance

    The great-circle distance is the shortest distance between any two Point s on the surface of a sphere measured along a path on the surface of the sphere ....
  • Spherical polyhedron
    Spherical polyhedron

    In mathematics, the surface of a sphere may be divided by line segments into bounded regions, to form a spherical tessellation or spherical polyhedron....
  • Spherical trigonometry
    Spherical trigonometry

    Spherical trigonometry is a part of spherical geometry that deals with polygons on the sphere and explains how to find relations between the involved angles....


External links

  • UNCC
  • Rice University
    Rice University

    William Marsh Rice University is a private university research university located in Houston, Texas, Texas, United States. The campus is located near the Houston Museum District and adjacent to the Texas Medical Center....