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Spherical geometry orspherics (from Ancient Greek σφαιρικά) is thegeometry of the two-dimensional surface of asphere[a] or then-dimensional surface ofhigher dimensional spheres.
Long studied for its practical applications toastronomy,navigation, andgeodesy, spherical geometry and the metrical tools ofspherical trigonometry are in many respects analogous toEuclidean plane geometry andtrigonometry, but also have some important differences.
The sphere can be studied eitherextrinsically as a surface embedded in 3-dimensionalEuclidean space (part of the study ofsolid geometry), orintrinsically using methods that only involve the surface itself without reference to any surrounding space.
Inplane (Euclidean) geometry, the basic concepts arepoints and (straight)lines. In spherical geometry, the basic concepts are points andgreat circles. However, two great circles on a plane intersect in two antipodal points, unlike coplanar lines inelliptic 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 ageodesic; 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 geometrybuilt 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 aspherical trigonometry that differs from ordinarytrigonometry in many respects; for example, the sum of the interior angles of a sphericaltriangle exceeds 180 degrees.
Because a sphere and a plane differ geometrically, (intrinsic) spherical geometry has some features of anon-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 theparallel postulate is a logical consequence of the rest of Euclid's axioms of plane geometry, because it requires another axiom to be modified. The resolution was found instead inelliptic geometry, to which spherical geometry is closely related, andhyperbolic geometry; each of these new geometries makes a different change to the parallel postulate.
The principles of any of these geometries can be extended to any number of dimensions.
An important geometry related to that of the sphere is that of thereal projective plane; it is obtained by identifyingantipodal 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 isnon-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 theoblong sphere, though minor modifications must be implemented on certain formulas.
The earliest mathematical work of antiquity to come down to our time isOn the rotating sphere (Περὶ κινουμένης σφαίρας,Peri kinoumenes sphairas) byAutolycus of Pitane, who lived at the end of the fourth century BC.[1]
Spherical trigonometry was studied by earlyGreek mathematicians such asTheodosius of Bithynia, a Greek astronomer and mathematician who wroteSpherics, a book on the geometry of the sphere,[2] andMenelaus of Alexandria, who wrote a book on spherical trigonometry calledSphaerica and developedMenelaus' theorem.[3][4]
The Book of Unknown Arcs of a Sphere written by the Islamic mathematicianAl-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 bookOn Triangles byRegiomontanus, 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 theAndalusi scholarJabir ibn Aflah.[6]
Leonhard Euler published a series of important memoirs on spherical geometry:
Spherical geometry has the following properties:[7]
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:
If "line" is taken to mean great circle, spherical geometry only obeys two of Euclid's five 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 thefifth (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 is180°(1 + 4f), wheref is the fraction of the sphere's surface that is enclosed by the triangle. For any positive value off, this exceeds 180°.
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