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**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 sphere" are used for the surface together with its 3-dimensional interior.

- Overview
- Relation to similar geometries
- History
- Greek antiquity
- Islamic world
- Euler's work
- Properties
- Relation to Euclid's postulates
- See also
- Notes
- References
- External links

Long studied for its practical applications to navigation and astronomy, 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), 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 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, angles 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 exceeds 180 degrees.

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 points (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.

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.^{ [1] }

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,^{ [2] } and Menelaus of Alexandria, who wrote a book on spherical trigonometry called *Sphaerica* and developed Menelaus' theorem.^{ [3] }^{ [4] }

*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.^{ [5] }

The book *On Triangles* by Regiomontanus, written around 1463, is the first pure trigonometrical work in Europe. However, Gerolamo Cardano 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.^{ [6] }

Leonhard Euler 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.

Spherical geometry has the following properties:^{ [7] }

- 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).

If "line" is taken to mean great circle, spherical geometry obeys two of Euclid's postulates: the second postulate ("to produce [extend] 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 points 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.^{ [8] }

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 180°(1 + 4*f*), 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°.

- ↑ Rosenfeld, B.A (1988).
*A history of non-Euclidean geometry : evolution of the concept of a geometric space*. New York: Springer-Verlag. p. 2. ISBN 0-387-96458-4. - ↑ "Theodosius of Bithynia – Dictionary definition of Theodosius of Bithynia".
*HighBeam Research*. Retrieved 25 March 2015. - ↑ O'Connor, John J.; Robertson, Edmund F., "Menelaus of Alexandria",
*MacTutor History of Mathematics archive*, University of St Andrews - ↑ "Menelaus of Alexandria Facts, information, pictures".
*HighBeam Research*. Retrieved 25 March 2015. - ↑ School of Mathematical and Computational Sciences University of St Andrews
- ↑ Victor J. Katz-Princeton University Press
- ↑ Merserve , pp. 281-282
- ↑ Gowers, Timothy,
*Mathematics: A Very Short Introduction*, Oxford University Press, 2002: pp. 94 and 98.

**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 (postulates), and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier, Euclid was the first to organize these propositions into a logical system in which each result is *proved* from axioms and previously proved theorems.

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 *r* from a given point in three-dimensional space. That given point is the centre of the sphere, and *r* is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.

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

A **great circle**, also known as an **orthodrome**, of a sphere is the intersection of the sphere and a plane that passes through the center point of the sphere. A great circle is the largest circle that can be drawn on any given sphere. Any diameter of any great circle coincides with a diameter of the sphere, and therefore all great circles have the same center and circumference as each other. This special case of a circle of a sphere is in opposition to a *small circle*, that is, the intersection of the sphere and a plane that does not pass through the center. Every circle in Euclidean 3-space is a great circle of exactly one sphere.

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 geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called non-Euclidean geometry.

The **Gauss–Bonnet theorem**, or **Gauss–Bonnet formula**, is a relationship between surfaces in differential geometry. It connects the curvature of a surface to its Euler characteristic.

**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 are usually assumed to intersect at a single point. Because of this, the elliptic geometry described in this article is sometimes referred to as *single elliptic geometry* whereas spherical geometry is sometimes referred to as *double elliptic geometry*.

In mathematics, **hyperbolic geometry** is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

The **great-circle distance**, **orthodromic distance**, or **spherical distance** is the distance along a great circle.

In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold. The various **charts on SO(3)** set up rival coordinate systems: in this case there cannot be said to be a preferred set of parameters describing a rotation. There are three degrees of freedom, so that the dimension of SO(3) is three. In numerous applications one or other coordinate system is used, and the question arises how to convert from a given system to another.

In geometry, the (**angular**) **defect** means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would. The opposite notion is the excess.

In a Euclidean space, the **sum of angles of a triangle** equals the straight angle . A triangle has three angles, one at each vertex, bounded by a pair of adjacent sides.

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 visualised in elliptic space.

**Trigonometry** is a branch of mathematics that studies the relationships between the sides and the angles in triangles. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves.

Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics and Babylonian mathematics. Trigonometry was also prevalent in Kushite mathematics. Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy. In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata, who discovered the sine function. During the Middle Ages, the study of trigonometry continued in Islamic mathematics, by mathematicians such as Al-Khwarizmi and Abu al-Wafa. It became an independent discipline in the Islamic world, where all six trigonometric functions were known. Translations of Arabic and Greek texts led to trigonometry being adopted as a subject in the Latin West beginning in the Renaissance with Regiomontanus. The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics and reaching its modern form with Leonhard Euler (1748).

Ordinary trigonometry studies triangles in the Euclidean plane . There are a number of ways of defining the ordinary Euclidean geometric trigonometric functions on real numbers, for example right-angled triangle definitions, unit circle definitions, series definitions, definitions via differential equations, and definitions using functional equations. **Generalizations of trigonometric functions** are often developed by starting with one of the above methods and adapting it to a situation other than the real numbers of Euclidean geometry. Generally, trigonometry can be the study of triples of points in any kind of geometry or space. A triangle is the polygon with the smallest number of vertices, so one direction to generalize is to study higher-dimensional analogs of angles and polygons: solid angles and polytopes such as tetrahedrons and n-simplices.

In mathematics, a **unit circle** is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as *S*^{1} because it is a one-dimensional unit *n*-sphere.

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 in the plane, and as in the planar case, a spherical conic can be defined as the locus of points the sum or difference of whose great-circle distances to two foci is constant. By taking the antipodal point to one focus, every **spherical ellipse** is also a **spherical hyperbola**, and vice versa. As a space curve, a spherical conic is a quartic, though its orthogonal projections in three principal axes are planar conics. Like planar conics, spherical conics also satisfy a “reflection property”: the great-circle arcs from the two foci to any point on the conic have the tangent and normal to the conic at that point as their angle bisectors.

- Meserve, Bruce E. (1983) [1959],
*Fundamental Concepts of Geometry*, Dover, ISBN 0-486-63415-9 - Papadopoulos, Athanase (2015),
*Euler, la géométrie sphérique et le calcul des variations. In: Leonhard Euler : Mathématicien, physicien et théoricien de la musique (dir. X. Hascher et A. Papadopoulos)*, CNRS Editions, Paris, ISBN 978-2-271-08331-9 - Van Brummelen, Glen (2013).
*Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry*. Princeton University Press. ISBN 9780691148922 . Retrieved 31 December 2014. - 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 ISBN 978-3-11-057142-4

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