Spherical geometry
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300px, A sphere with a spherical triangle on it. Spherical geometry is the
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
of the two-
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
al surface of a
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sphere" are used for the surface together with its 3-dimensional interior. Long studied for its practical applications to
navigation Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, ...
and
astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
, spherical geometry bears many similarities and relationships to, and important differences from, Euclidean
plane geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
. The sphere has for the most part been studied as a part of 3-dimensional Euclidean geometry (often called
solid geometry In mathematics, solid geometry or stereometry is the traditional name for the geometry of three-dimensional, Euclidean spaces (i.e., 3D geometry). Stereometry deals with the measurements of volumes of various solid figures (or 3D figures), inc ...
), the surface thought of as placed inside an ambient 3-d space. It can also be analyzed by "intrinsic" methods that only involve the surface itself, and do not refer to, or even assume the existence of, any surrounding space outside or inside the sphere. Because a sphere and a plane differ geometrically, (intrinsic) spherical geometry has some features of a
non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
and is sometimes described as being one. However, spherical geometry was not considered a full-fledged non-Euclidean geometry sufficient to resolve the ancient problem of whether the
parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ''If a line segment ...
is a logical consequence of the rest of Euclid's axioms of plane geometry. The solution was found instead in
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...
.


Overview

In plane (Euclidean) geometry, the basic concepts are
point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...
s and (straight) lines. In spherical geometry, the basic concepts are point and
great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...
. However, two great circles on a plane intersect in two antipodal points, unlike coplanar lines in
Elliptic geometry Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines ...
. In the extrinsic 3-dimensional picture, a great circle is the intersection of the sphere with any plane through the center. In the intrinsic approach, a great circle is a
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
; a shortest path between any two of its points provided they are close enough. Or, in the (also intrinsic) axiomatic approach analogous to Euclid's axioms of plane geometry, "great circle" is simply an undefined term, together with postulates stipulating the basic relationships between great circles and the also-undefined "points". This is the same as Euclid's method of treating point and line as undefined primitive notions and axiomatizing their relationships. Great circles in many ways play the same logical role in spherical geometry as lines in Euclidean geometry, e.g., as the sides of (spherical) triangles. This is more than an analogy; spherical and plane geometry and others can all be unified under the umbrella of geometry built from distance measurement, where "lines" are defined to mean shortest paths (geodesics). Many statements about the geometry of points and such "lines" are equally true in all those geometries provided lines are defined that way, and the theory can be readily extended to higher dimensions. Nevertheless, because its applications and pedagogy are tied to solid geometry, and because the generalization loses some important properties of lines in the plane, spherical geometry ordinarily does not use the term "line" at all to refer to anything on the sphere itself. If developed as a part of solid geometry, use is made of points, straight lines and planes (in the Euclidean sense) in the surrounding space. In spherical geometry,
angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...
s are defined between great circles, resulting in a
spherical trigonometry 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 gr ...
that differs from ordinary
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
in many respects; for example, the sum of the interior angles of a spherical
triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...
exceeds 180 degrees.


Relation to similar geometries

Spherical geometry is closely related to
elliptic geometry Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines ...
. An important geometry related to that of the sphere is that of the
real projective plane In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
; it is obtained by identifying
antipodal point 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 d ...
s (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 In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
, or one-sided, and unlike the sphere it cannot be drawn as a surface in 3-dimensional space without intersecting itself. 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 is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines ...
.


History


Greek antiquity

The earliest mathematical work of antiquity to come down to our time is ''On the rotating sphere'' (Περὶ κινουμένης σφαίρας, ''Peri kinoumenes sphairas'') by
Autolycus of Pitane Autolycus of Pitane ( el, Αὐτόλυκος ὁ Πιταναῖος; c. 360 – c. 290 BC) was a Greek astronomer, mathematician, and geographer. The lunar crater Autolycus was named in his honour. Life and work Autolycus was born in Pitane, ...
, who lived at the end of the fourth century BC. Spherical trigonometry was studied by early
Greek mathematicians Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
such as
Theodosius of Bithynia Theodosius of Bithynia ( grc-gre, Θεοδόσιος; c. 169 BCc. 100 BC) was a Greek astronomer and mathematician who wrote the ''Sphaerics'', a book on the geometry of the sphere. Life Born in Tripolis, in Bithynia, Theodosius was m ...
, a Greek astronomer and mathematician who wrote the
Sphaerics ''Sphaerics'' ( grc, Σφαιρικά) was a set of three volumes on spherical geometry written by Theodosius of Bithynia in the 2nd century BC. These proved essential in the restoration of Euclidean geometry to Western civilization, when brought ...
, a book on the geometry of the sphere, and
Menelaus of Alexandria Menelaus of Alexandria (; grc-gre, Μενέλαος ὁ Ἀλεξανδρεύς, ''Menelaos ho Alexandreus''; c. 70 – 140 CE) was a GreekEncyclopædia Britannica "Greek mathematician and astronomer who first conceived and defined a spheric ...
, who wrote a book on spherical trigonometry called ''Sphaerica'' and developed
Menelaus' theorem Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle ''ABC'', and a transversal line that crosses ''BC'', ''AC'', and ''AB'' at points ''D'', ''E'', and ''F'' respec ...
.


Islamic world

''The Book of Unknown Arcs of a Sphere'' written by the Islamic mathematician
Al-Jayyani Abū ʿAbd Allāh Muḥammad ibn Muʿādh al-Jayyānī ( ar, أبو عبد الله محمد بن معاذ الجياني; 989, Cordova, Al-Andalus – 1079, Jaén, Al-Andalus) was an Arab, mathematician, Islamic scholar, and Qadi from Al-Anda ...
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 Johannes Müller von Königsberg (6 June 1436 – 6 July 1476), better known as Regiomontanus (), was a mathematician, astrologer and astronomer of the German Renaissance, active in Vienna, Buda and Nuremberg. His contributions were instrument ...
, written around 1463, is the first pure trigonometrical work in Europe. However,
Gerolamo Cardano Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...
noted a century later that much of its material on spherical trigonometry was taken from the twelfth-century work of the
Andalusi The Arabic '' nisbah'' (attributive title) Al-Andalusi denotes an origin from Al-Andalus. Al-Andalusi may refer to: * Abu Hayyan al-Gharnati * Ibn Hazm * Ibn Juzayy * Ibn 'Atiyya * Said Al-Andalusi Ṣāʿid al-Andalusī (); he was Abū al-Qāsi ...
scholar
Jabir ibn Aflah Abū Muḥammad Jābir ibn Aflaḥ ( ar, أبو محمد جابر بن أفلح, la, Geber/Gebir; 1100–1150) was an Arab Muslim astronomer and mathematician from Seville, who was active in 12th century al-Andalus. His work ''Iṣlāḥ al-Ma ...
.


Euler's work

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
published a series of important memoirs on spherical geometry: * L. Euler, Principes de la trigonométrie sphérique tirés de la méthode des plus grands et des plus petits, Mémoires de l'Académie des Sciences de Berlin 9 (1753), 1755, p. 233–257; Opera Omnia, Series 1, vol. XXVII, p. 277–308. * L. Euler, Eléments de la trigonométrie sphéroïdique tirés de la méthode des plus grands et des plus petits, Mémoires de l'Académie des Sciences de Berlin 9 (1754), 1755, p. 258–293; Opera Omnia, Series 1, vol. XXVII, p. 309–339. * L. Euler, De curva rectificabili in superficie sphaerica, Novi Commentarii academiae scientiarum Petropolitanae 15, 1771, pp. 195–216; Opera Omnia, Series 1, Volume 28, pp. 142–160. * L. Euler, De mensura angulorum solidorum, Acta academiae scientiarum imperialis Petropolitinae 2, 1781, p. 31–54; Opera Omnia, Series 1, vol. XXVI, p. 204–223. * L. Euler, Problematis cuiusdam Pappi Alexandrini constructio, Acta academiae scientiarum imperialis Petropolitinae 4, 1783, p. 91–96; Opera Omnia, Series 1, vol. XXVI, p. 237–242. * L. Euler, Geometrica et sphaerica quaedam, Mémoires de l'Académie des Sciences de Saint-Pétersbourg 5, 1815, p. 96–114; Opera Omnia, Series 1, vol. XXVI, p. 344–358. * L. Euler, Trigonometria sphaerica universa, ex primis principiis breviter et dilucide derivata, Acta academiae scientiarum imperialis Petropolitinae 3, 1782, p. 72–86; Opera Omnia, Series 1, vol. XXVI, p. 224–236. * L. Euler, Variae speculationes super area triangulorum sphaericorum, Nova Acta academiae scientiarum imperialis Petropolitinae 10, 1797, p. 47–62; Opera Omnia, Series 1, vol. XXIX, p. 253–266.


Properties

Spherical geometry has the following properties: * Any two great circles intersect in two diametrically opposite points, called ''antipodal points''. * Any two points that are not antipodal points determine a unique great circle. * There is a natural unit of angle measurement (based on a revolution), a natural unit of length (based on the circumference of a great circle) and a natural unit of area (based on the area of the sphere). * Each great circle is associated with a pair of antipodal points, called its ''poles'' which are the common intersections of the set of great circles perpendicular to it. This shows that a great circle is, with respect to distance measurement ''on the surface of the sphere'', a circle: the locus of points all at a specific distance from a center. * Each point is associated with a unique great circle, called the ''polar circle'' of the point, which is the great circle on the plane through the centre of the sphere and perpendicular to the diameter of the sphere through the given point. As there are two arcs determined by a pair of points, which are not antipodal, on the great circle they determine, three non-collinear points do not determine a unique triangle. However, if we only consider triangles whose sides are minor arcs of great circles, we have the following properties: * The angle sum of a triangle is greater than 180° and less than 540°. * The area of a triangle is proportional to the excess of its angle sum over 180°. * Two triangles with the same angle sum are equal in area. * There is an upper bound for the area of triangles. * The composition (product) of two reflections-across-a-great-circle may be considered as a rotation about either of the points of intersection of their axes. * Two triangles are congruent if and only if they correspond under a finite product of such reflections. * Two triangles with corresponding angles equal are congruent (i.e., all similar triangles are congruent).


Relation to Euclid's postulates

If "line" is taken to mean great circle, spherical geometry obeys two of Euclid's postulates: the second postulate ("to produce
xtend Xtend is a general-purpose high-level programming language for the Java Virtual Machine. Syntactically and semantically Xtend has its roots in the Java programming language but focuses on a more concise syntax and some additional functionality ...
a finite straight line continuously in a straight line") and the fourth postulate ("that all right angles are equal to one another"). However, it violates the other three. Contrary to the first postulate ("that between any two points, there is a unique line segment joining them"), there is not a unique shortest route between any two points (
antipodal point 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 d ...
s such as the north and south poles on a spherical globe are counterexamples); contrary to the third postulate, a sphere does not contain circles of arbitrarily great radius; and contrary to the fifth (parallel) postulate, there is no point through which a line can be drawn that never intersects a given line. Gowers, Timothy, ''Mathematics: A Very Short Introduction'', Oxford University Press, 2002: pp. 94 and 98. A statement that is equivalent to the parallel postulate is that there exists a triangle whose angles add up to 180°. Since spherical geometry violates the parallel postulate, there exists no such triangle on the surface of a sphere. The sum of the angles of a triangle on a sphere is , where ''f'' is the fraction of the sphere's surface that is enclosed by the triangle. For any positive value of ''f'', this exceeds 180°.


See also

*
Spherical astronomy Spherical astronomy, or positional astronomy, is a branch of observational astronomy used to locate astronomical objects on the celestial sphere, as seen at a particular date, time, and location on Earth. It relies on the mathematical methods o ...
*
Spherical conic In mathematics, a spherical conic or sphero-conic is a curve on the sphere, the intersection of the sphere with a concentric elliptic cone. It is the spherical analog of a conic section (ellipse, parabola, or hyperbola) in the plane, and as in th ...
*
Spherical distance The great-circle distance, orthodromic distance, or spherical distance is the distance along a great circle. It is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a st ...
*
Spherical polyhedron In geometry, a spherical polyhedron or spherical tiling is a tessellation, tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhe ...
*
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. Fo ...
*
Lénárt sphere A Lénárt sphere is a educational manipulative and writing surface for exploring spherical geometry, invented by Hungarian István Lénárt as a modern replacement for a spherical blackboard. It can be used for visualizing the geometry of po ...
*
Versor In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Will ...


Notes


References

* * * *
Roshdi Rashed Roshdi Rashed (Arabic: رشدي راشد), born in Cairo in 1936, is a mathematician, philosopher and historian of science, whose work focuses largely on mathematics and physics of the medieval Arab world. His work explores and illuminates the un ...
and Athanase Papadopoulos (2017) ''Menelaus' Spherics: Early Translation and al-Mahani'/alHarawi's version. Critical edition of Menelaus' Spherics from the Arabic manuscripts, with historical and mathematical commentaries'',
De Gruyter Walter de Gruyter GmbH, known as De Gruyter (), is a German scholarly publishing house specializing in academic literature. History The roots of the company go back to 1749 when Frederick the Great granted the Königliche Realschule in Be ...
Series: Scientia Graeco-Arabica 21


External links


The Geometry of the Sphere
Rice University William Marsh Rice University (Rice University) is a Private university, private research university in Houston, Houston, Texas. It is on a 300-acre campus near the Houston Museum District and adjacent to the Texas Medical Center. Rice is ranke ...
*
Navigation Spreadsheets: Navigation TrianglesSphaerica - geometry software for constructing on the sphere
{{Authority control Classical geometry Spherical trigonometry Spherical astronomy