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 mathematical 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 (1D) because only one coor ...
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. 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 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 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 points and (straight) lines. In spherical geometry, the basic concepts are point and great circle. However, two great circles on a plane intersect in two antipodal points, unlike coplanar lines in Elliptic geometry. 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; 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 rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
s are defined between great circles, resulting in a spherical trigonometry that differs from ordinary trigonometry in many respects; for example, the sum of the interior angles of a spherical
triangle A triangle is a polygon with three edges and three 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, any three points, when non- colli ...
exceeds 180 degrees.


Relation to similar geometries

Spherical geometry is closely related to elliptic geometry. An important geometry related to that of the sphere is that of the real projective plane; 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 ...
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, 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.


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, who lived at the end of the fourth century BC. Spherical trigonometry was studied by early Greek mathematicians such as Theodosius of Bithynia, a Greek astronomer and mathematician who wrote the Sphaerics, 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 Greek Encyclopædia Britannica "Greek mathematician and astronomer who first conceived and defined a sphe ...
, who wrote a book on spherical trigonometry called ''Sphaerica'' and developed Menelaus' theorem.


Islamic world

''The Book of Unknown Arcs of a Sphere'' written by the Islamic mathematician Al-Jayyani 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, 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 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 a ...
.


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 xtenda 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 ...
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 conic * Spherical distance *
Spherical polyhedron In geometry, a spherical polyhedron or spherical tiling is a 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 polyhedra is most c ...
* Half-side formula * Lénárt sphere * Versor


Notes


References

* * * * Roshdi Rashed 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 Series: Scientia Graeco-Arabica 21


External links


The Geometry of the Sphere
Rice University *
Navigation Spreadsheets: Navigation TrianglesSphaerica - geometry software for constructing on the sphere
{{Authority control Classical geometry Spherical trigonometry Spherical astronomy